I'm looking at the free energy of a body (theory of elasticity) but I can't really square the general expression with the one usually used for isotropic bodies. According to wikipedia (http://en.wikipedia.org/wiki/Elastic_energy), Landau & Lifgarbagez etc the general expression for the free (elasticity) energy "F" of a body is ½*Cijkl*uij*ukl (summation implied) where Cijkl is the elastic modulus tensor and the uij's are the strain tensor, uij=½*((∂ui/∂xj)+(∂uj/∂xi))=uji (small deformations, i.e. no non-linear term). Considering 2D, the non-zero components of Cijkl for an isotropic medium (as can be seen e.g. on the above wikipedia page) are Cxxxx=Cyyyy=λ+2μ Cxyxy=Cyxyx=Cxyyx=Cyxxy=μ Cxxyy=Cyyxx=λ where λ and μ are the Lame coefficients (material constants). Now, if I use these above values, I get (½λ+2μ)*[uxx2+uyy2]+2μ*uxy2+λ*uxx*uyy . According to Landau & Lifgarbagez, a million papers etc, the free energy of an isotropic body is F=½λ*uii2+μ*uik2=(½λ+μ)*[uxx2+uyy2]+2μ*uxy2 . I would expect the general expression to reduce to the isotropic one when using the isotropic elastic modulus tensor. However, I get an extra term "+λ*uxx*uyy" which does not appear in the usual isotropic expression. Moreover, this extra term seems quite unphysical too me, since it does not involve a square, and thus could probably be negative for a non-zero deformation. It seems to me that this term should go, however, I do not see how.