Towards Analogue Transformation Elasticity Transformation techniques rely on forminvariance
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1 Towards Analogue Transformation Elasticity Transformation techniques rely on forminvariance of field equations Elasticity: standard eqs. can be written in spatial tensor form (i.e. form-invariant) What about space-time mixing transformations? Could Analogue Transformation techniques be relevant for Elasticity? What about composite materials? C. García Meca, S. Carloni, C. Barceló, G. Jannes, J. Sánchez-Dehesa, and A. Martínez ACT ARIADNA PROJECT Technical Report
2 Disclaimer Field of Transformation Elasticity is little developed No off-the-shelf analogue relativistic model for Elasticity Preliminary study of fundamental issues
3 Standard linear elasticity Elastic wave equation (standard textbook eq.): C: elasticity tensor u: displacement vector ρ: density Cartesian coordinates! Purely spatial! Can this be written tensorially? YES follows from (spatially) tensorial character of basic eqs. of linear elasticity
4 Space-time transformations: abstract relativistic system Acoustics: velocity potential - scalar eq. Elasticity: vector equation (displ. u) (much) more complicated, e.g. spatial elasticity tensor C must be generalized to spacetime tensor: C ijkl C μνρσ ; also u i u μ
5 Space-time transformations: abstract relativistic system Acoustics: velocity potential - scalar eq. Elasticity: vector equation (displ. u) (much) more complicated, e.g. spatial elasticity tensor C must be generalized to spacetime tensor: C ijkl C μνρσ ; also u i u μ how to fix u 0? f(u i ) non-linearities? spacetime mixing terms: background flows? relation to material parameters of some familiar type of elasticity?
6 More exotic transformation laws (1) E.g. choose spatial transf. matrix A freely independently of coord. change matrix [Norris-Shuvalov 2011] spoils form-invariance could be interesting w.r.t. peculiar types of (composite) elastic materials within AT framework: choose non-tensorial abstract intermediate system
7 More exotic transformation laws (1) E.g. choose spatial transf. matrix A freely independently of coord. change matrix [Norris-Shuvalov 2011] spoils form-invariance could be interesting w.r.t. peculiar types of (composite) elastic materials within AT framework: choose non-tensorial abstract intermediate system E.g. Willis equations : can be realized in elastic composite structures and random media [Milton, Briane, Willis 2006] + : very general eqs.; closed under transformations - : formal character (no exact expressions for S eff,c eff,ρ eff ) Periodically structured materials in long λ limit: usual eqs of elasticity (in terms of avgd values)
8 More exotic transformation laws (2) E.g. space-time transformations: take u as spatial vector + time scalar (i.e. u 0 =0) spoils form-invariance but could avoid nonlinearities
9 More exotic transformation laws (2) E.g. space-time transformations: take u as spatial vector + time scalar (i.e. u 0 =0) spoils form-invariance but could avoid nonlinearities In both examples, crucial question (again): connection with familiar elastic systems? prospects of realizing it in laboratory?
10 Conclusions Scope of transformational elasticity very broad Different tensorial transformation laws (3D, 4D) Non-tensorial transformation laws Still in need of systematic treatment of essentials of transformation elasticity far beyond clean form-invariance optics/acoustics Essential to focus on specific examples of elastic systems desirability + realizability Spacetime transformations will require temporal control of metamaterial properties to achieve (apparent) flows (homogenization?)
11 Conclusions Scope of transformational elasticity very broad Different tensorial transformation laws (3D, 4D) Non-tensorial transformation laws Still in need of systematic treatment of essentials of transformation elasticity far beyond clean form-invariance optics/acoustics Essential to focus on specific examples of elastic systems desirability + realizability Spacetime transformations will require temporal control of metamaterial properties to achieve (apparent) flows (homogenization?) WORK IN PROGRESS!
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