Keywords: electric-thermal analysis, link finite element, functionally graded material.

The theoretical part of this contribution deals with deriving the effective matrices of the new electric-thermal functionally graded material (FGM) link finite element of constant cross-sectional area with spatial continuous variation of electric and thermal material properties. This variation of the effective material properties can be caused by both continuous longitudinal and continuous symmetric transversal variation of the constituent's volume fractions and constituent's material properties. All the variation in the material properties is included in the element matrices very accurately through transfer functions and transfer constants [1,2]. This concept enables a very accurate distribution of the calculated variables in the conductor field for very coarse finite element mesh to be obtained. Homogenization of the material properties (the reference volume is the volume of the whole link finite element) will be done in two steps. In the first step, the real link element will be transformed to a multilayered link element. Material properties of the layers will be calculated with the extended mixture rules [1]. Each layer will have constant volume fractions and material properties of the constituents through the link height and width. They are calculated as an average value from their values on the boundaries of the respective layer. Polynomial variation of these parameters will appear in the longitudinal direction. Sufficient accuracy of substitution of the continuous lateral variation of material properties by the layer-wise constant lateral distribution of material properties will be reached when the division to layers is fine enough. In the second step, by using the laminate theory [1,3], the effective longitudinal material properties of the homogenized link element will be derived. These homogenized effective material properties (electrical and thermal conductance) are constant through the beam height and width, but they vary continuously along the longitudinal link axis. The element equation is then established for the homogenized link finite element for calculation of the effective link conductor variables.

The contribution of the distributed Joule's heat along the element length on the temperature rise of the electric conductor is considered. Numerical experiments are done concerning the analyses of a one-dimensional electric conductor. The effect of the division fineness of the line conductor to the layers in the transverse direction on the solution accuracy is evaluated. The solution results are discussed and compared with those obtained using a very fine mesh of conventional finite elements.

1

J. Murín, V. Kutiš, "An effective multilayered sandwich beam-link finite element for solution of the elecro-thermo-structural problems", Int. J. Computers and Structures, 87, 1496-1507, 2009. doi:10.1016/j.compstruc.2009.06.009

2

H. Rubin, "Analytische Lösung linearer Differentialgleichungen mit veränderlichen Koeffizienten und baustatische Anwendung", Bautechnik, 76, 1999.

3

H. Altenbach, J. Altenbach, W. Kissing, "Mechanics of Composite Structural Elements", Engineering - Monograph (English), Springer-Verlag, 2003.

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