Given a 4th order tensor and its Cartesian components, and a convected (non-orthogonal) covariant basis, how can I calculate variously the covariant and contravariant components of the tensor?
Specifically, I have an elasticity tensor.
The Cartesian basis is: $\vec{e_1}$ = (1,0,0), $\vec{e_2}$ = (0,1,0), $\vec{e_3}$ = (0,0,1).
The convected basis is: $\vec{g_1}$ = (-1,-0.5,0), $\vec{g_2}$ = (-1,0.5,0), $\vec{g_3}$ = (0,0,-1/400).
The elasticity tensor $D_{ijkl}$ has the usual 81 components.
I am particularly interested in the contravariant tensor components in the convected basis, i.e. $\tilde{D}^{mnop}$.
As a first step, I've calculated the dual (contravariant) basis: $\vec{g^1}$ = (-0.5,-1,0), $\vec{g^2}$ = (-0.5,1,0), $\vec{g^3}$ = (0,0,-400).
How should I proceed from this point?
The procedure would be:
1) First find $\widetilde{D}_{mnop}$ in terms of $D_{ijk\ell}$. Here's how: if $\vec{g}_j = a^i_{~j}\vec{e}_i$, then $\widetilde{D}_{mnop} = a^i_{~m}a^{j}_{~ n}a^k_{~o}a^\ell_{~p}D_{ijk\ell}$, by multilinearity of $D$.
2) Raise all the indices by using the expression for the metric in terms of the convected basis, as follows: let $g_{ij} = \vec{g}_i \cdot \vec{g}_j$. This matrix is not the identity, as the convected basis is not orthonormal. Let $(g^{ij})$ be the inverse matrix. Then $\widetilde{D}^{mnop} = g^{mm'} g^{nn'} g^{oo'} g^{pp'}\widetilde{D}_{m'n'o'p'}$.