Mechanics of Continua and Structures
The relative position vectors are defined as follows: Π(X)=X−ˉX,πt(x)=x−ˉxt,
The Euler tensors E0 and E are defined as follows:
E0=∫Ω0Π⊗Πρ0dΩ0Et=∫Ωtπ⊗πρtdΩtThe inertia tensors J0 and Jt are defined as follows:
J0=tr(E0)I−E0,Jt=tr(Et)I−Et,Ellipsoid example: Calculation in Mathematica
The spatial tensors Et and Jt can be computed from their material counterparts. Specifically, it can be shown (part of your HW 5 prblem set) that,
Et=RtE0RTtJt=RtJ0RTtLet ˆEi be Cartesian basis vectors. Say that (J0)ij (resp. (E0)ij) are the components of J (resp. E0) w.r.t. ˆEi, i.e.,
(J0)ij=ˆEi⋅(J0ˆEj)(E0)ij=ˆEi⋅(E0ˆEj)Following the dyadic representation of tensors, we can also write J0=(J0)ijˆEi⊗ˆEjE0=(E0)ijˆEi⊗ˆEj
Say ˆei are corresponding co-rotational basis vectors corresponding to ˆEi. It can be shown that Jt=(J0)ijˆei⊗ˆejEt=(E0)ijˆei⊗ˆej
Proof. Let Ft stand for either Et or Jt, and Let F0 stand for either E0 or J0.
Ft=RtF0RTt=Rt((F0)ijˆEi⊗ˆEj)RTt=(F0)ijRt(ˆEi⊗ˆEj)RTt=(F0)ijRt(ˆEi⊗((RTt)TˆEj))=(F0)ijRt(ˆEi⊗(RtˆEj))=(F0)ij((RtˆEi)⊗(RtˆEj))=(F0)ij(RtˆEi)⊗(RtˆEj)=(F0)ij(ˆei)⊗(ˆej)=(F0)ijˆei⊗ˆej