On the Calculation of the Eshelby Tensor and the Beauty of our Nature
The calculation of the Eshelby tensor occupies the dominant part of time in modern selfconsistent schemes modeling texture developments. A further time-reduction can be got representing the denominator in the explicit expression of the integrand of the Eshelby integral by finite series, leading to known library integrals. The possibility to get fast converging series depends on the degree of anisotropy of the stiffness tensor C. As universal (any symmetries) parameter, in order to classify the degree of anisotropy of a given C-data set, the variance s0 of the normalized determinant D0(C,r) of the Christoffel matrix is suggested. s0 can exactly be determined without great effort. For typical C-sets time reducing factors in the order of 2 – 8 have been got. Beyond this special result the anisotropy parameter 0 £ s0 < 1 seems to be of general importance. 3d-figures of D0(C,r) show characteristic (and sometimes unexpected ‘beautiful’) individual faces of the elastic properties of monocrystalline or textured polycrystalline substances.
C. Esling, M. Humbert, R.A. Schwarzer and F. Wagner
S. Matthies and G.W. Vinel, "On the Calculation of the Eshelby Tensor and the Beauty of our Nature", Solid State Phenomena, Vol. 105, pp. 113-120, 2005