Reduced-Strain Closure Model

As experimental evidence has shown that the standard Folgar-Tucker model predicts a faster transient orientation evolution than that observed experimentally. Tucker et al. (2007), Wang et al. (2008) and Phelps and Tucker (2009) have proposed a new evolution equation, i.e., the so-called reduced-strain closure (RSC) model, to slow down the fiber orientation kinetics. Their approach is based on the spectral decomposition theorem. The theorem states that if T is a symmetric second-order tensor, then there is a basis e, i — 1, 2, 3 consisting entirely of eigenvectors of T and the corresponding eigenvalues Aj, i — 1, 2, 3 forming the entire spectrum of T, thus T can be represented by T = A,e,e,. 

The theorem allows decomposition of the evolution equation for the second-order tensor into two rate equations for the eigenvalues and eigenvectors of ay, respectively. Then one modifies the equation for the eigenvalues, and keeps the equation for the eigenvectors unchanged. Finally, a new equation is formed by reassembling the equations. This approach preserves objectivity of the equation. For details about objectivity, see for example Tanner (2000). The resulting RSC equation is 

A similar model has also been proposed by Ferec et al. (2009) independently. 

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