Geometry and mechanics of two-dimensional defects in amorphous materials

Abstract

We study the geometry of defects in amorphous materials and their elastic interactions. Defects are defined and characterized by deviations of the material's intrinsic metric from a Euclidian metric. This characterization makes possible the identification of localized defects in amorphous materials, the formulation of a corresponding elastic problem, and its solution in various cases of physical interest. We present a multipole expansion that covers a large family of localized 2D defects. The dipole term, which represents a dislocation, is studied analytically and experimentally. Quadrupoles and higher multipoles correspond to fundamental strain-carrying entities. The interactions between those entities, as well as their interaction with external stress fields, are fundamental to the inelastic behavior of solids. We develop analytical tools to study those interactions. The model, methods, and results presented in this work are all relevant to the study of systems that involve a distribution of localized sources of strain. Examples are plasticity in amorphous materials and mechanical interactions between cells on a flexible substrate.amorphous solid | elasticity | reference metric | plasticity | defects

Renormalization of elastic quadrupoles in amorphous solids

“…In fact, all the terms in parentheses in Eqs. (21), (22) scale as a 0 , hence the flow is homogeneous, which implies that the initial value of a is forgotten and the universality hypothesis is verified. A key role is played by the single-defect partition function Z τ = τ e −βV (τ ) , which controls the dipole-moment scales and fluctuations appearing as τ 1 , a τ , τ 2 , and χ above.…”

“…First, we look for fixed points. Assuming A = 0 and χ = 0 as we will check later, the flow equation for γ 22 requires that γ 22 = 0. This then implies that all the other interactions are stationary, and only γ 12 can be nonzero.…”

Geometric charges and nonlinear elasticity of two-dimensional elastic metamaterials

“…Appendix B: Two types of multipoles Elastic charges are quantified by curvature singularity of a reference geometry [13,14]. Monopole charges correspond to delta-function singularity, leading to an equation of the form…”

“…There is another isotropic charge that is not conserved, but corresponds to local elastic deformations, that is the isotropic Eshelby inclusion [15]. Within the formalism of elastic charges this object is quantified as the Laplacian of delta function singularity [14] 1…”

Defect-Driven Shape Instabilities of Bundles

“…Complementary to this mechanical perspective is the geometrical one, in which topological defects redefine the metric of the 2D surface in which the crystal is embedded [4]. Consistent with the Gauss-Bonnet theorem [35], the deficit or excess angle associated with disclinations can be accommodated without far-field strain, provided that it is balanced by the integrated Gaussian curvature of the sheet.…”