Wärmeleitfähigkeitstensor

The Wärmeleitfähigkeitstensor, or thermal conductivity tensor in English, is a mathematical object used to describe the anisotropic thermal conductivity of a material. In many materials, heat flows equally well in all directions, and their thermal conductivity can be represented by a single scalar value. However, in anisotropic materials, the rate at which heat conducts depends on the direction. This directional dependence is captured by the thermal conductivity tensor.

This tensor is a second-order tensor, typically represented as a 3x3 matrix. The elements of this matrix, denoted by k_ij, relate the heat flux vector (the amount of heat energy flowing per unit area per unit time) to the temperature gradient vector (the rate of change of temperature with respect to position). Specifically, the relationship is given by Fourier's Law of Heat Conduction, expressed in tensor form: q = -K ∇T, where q is the heat flux vector, K is the thermal conductivity tensor, and ∇T is the temperature gradient.

The diagonal elements of the tensor (k_xx, k_yy, k_zz) represent the thermal conductivity along the principal axes of the material. The off-diagonal elements account for the coupling between heat flow in different directions. In many cases, the thermal conductivity tensor can be simplified by choosing a coordinate system aligned with the material's principal axes, where the off-diagonal elements become zero, and the tensor reduces to a diagonal matrix. This simplification highlights the distinct thermal conductivities along these principal directions. Understanding the Wärmeleitfähigkeitstensor is crucial for accurate thermal modeling and analysis of materials exhibiting anisotropic thermal properties, such as composites, certain crystals, and textured metals.