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Discrete Transformation Elastodynamics

Last modified: 2011-12-30

#### Abstract

A discrete version of transformation elastodynamics is introduced. This requires a new type of spring, which we call a ``torque spring'' where the forces at the end of the spring are equal and opposite but not directed in line with the spring. We show how torque springs can be constructed within the framework of linear elastodnamics, neglecting gravity and stability questions. The homogenization of a network of torque springs gives rise to metamaterials in which infinitesimal rotations cause stresses and stresses are not generally symmetric: the effective elasticity tensor does not satisfy the usual minor symmetries. We also construct a discrete model in which for linear elasticity the effective mass density is zero at all frequencies, but nonetheless the effective elasticity tensor is frequency dependent: the internal masses do not move when the material is translated but do move if the material is stretched. In any of these models the local displacement, and not just the local displacement gradient, needs to be small.

#### Keywords

Metamaterials, Transformation Elastodynamics