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In the evolution equation of the second order fiber orientation tensor (bar {boldsymbol {N}}), the fourth order fiber orientation tensor (bar {mathbb {N}}) appears.
Both of them depend linearly on the fourth order fiber orientation tensor.
Analogously, in the evolution equation of (bar {mathbb {N}}), the next higher even order fiber orientation tensor appears.
The initial fiber orientation state is isotropic, this is represented also in the second- and fourth order fiber orientation tensor component plots in Fig. 2 a and b.
The subfigures b, c and d represent the evolution of the components of the second order fiber orientation tensor (bar {boldsymbol {N}}) for the case of a shear flow, with identical isotropic initial conditions.
and it is ensured that the new second order fiber orientation tensor has unit trace bar{boldsymbol{N}}(t+Delta t) = frac {bar{boldsymbol{N}}(t) + dot{bar{boldsymbol{N}}}(t)Delta t} {text{tr}(bar{boldsymbol{N}}(t) + dot{bar{boldsymbol{N}}}(t)Delta t)}.
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The second step computes the thermal conductivity of the UD composite formed by the equivalent SiC fibers embedded in a SiC matrix, and finally the thermal conductivity of the as-formed SiC/SiC composite is obtained by averaging the solution for the UD composite over all possible fiber orientations using the second-order fiber orientation tensor.
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