Consider a hypothetical linear stress-strain relation for an isotropic solid,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids

Invariance, Ch Index notation and Cartesian tensors

Problem

\sigma_{ij}=C_{ijkl}e_{kl}

, where

e_{kl}=e_{lk}

is the (symmetric) strain. Someone proposes adding, to the ordinary Lamé terms, an isotropic antisymmetric piece

C^{\rm extra}_{ijkl}=\kappa\,(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})

for some scalar

\kappa

. ((a)) Using the contracted Levi-Civita identity in the form

\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk}=\epsilon_{ijm}\epsilon_{klm}

, write

C^{\rm extra}_{ijkl}e_{kl}

\kappa\,\epsilon_{ijm}w_m

for some vector

w_m

, and identify

w_m

in terms of

e_{kl}

. ((b)) Show that

w_m=0

for any symmetric strain, so that this extra term never contributes to the stress. ((c)) Say in one sentence why this is consistent with the ordinary reduction of the elastic tensor to the two Lamé constants

\lambda,\mu

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