An ideal nematic liquid crystal elastomer characterised by the strain-energy function W(nc) (F,n)W(A), where F represents...

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An ideal nematic liquid crystal elastomer characterised by the strain-energy function W(nc) (F,n)W(A), where F represents the macroscopic deformation gradient from the cross-linking state, n is the nematic director in the present configuration, W(A) is the strain-energy density of the isotropic polymer network, depending only on the local elastic deformation tensor A, and F = GAG, where Go and G are the natural deformation tensors due to the liquid crystal orientation in the reference and current configuration, respectively. Prove that, by the principle of material objectivity, the following tensor is symmetric. JW (nc) -FT F aw (nc) + n n An ideal nematic liquid crystal elastomer characterised by the strain-energy function W(nc) (F,n)W(A), where F represents the macroscopic deformation gradient from the cross-linking state, n is the nematic director in the present configuration, W(A) is the strain-energy density of the isotropic polymer network, depending only on the local elastic deformation tensor A, and F = GAG, where Go and G are the natural deformation tensors due to the liquid crystal orientation in the reference and current configuration, respectively. Prove that, by the principle of material objectivity, the following tensor is symmetric. JW (nc) -FT F aw (nc) + n n