Coupled Effects: Joule-Heating

Heterogeneous microstructures lead to a distortion of the electrical and current field within the material mixture. This leads to the fields becoming amplified within the material, which can lead to a variety of detrimental effects. An important quantity of interest is the amount of heat generated from an electrical field. The interconversions of various forms of energy (electromagnetic, thermal, etc.) in a system are governed by the first law of thermodynamics (which will be derived in detail shortly),

$\rho \dot{w}-{\bf T}:\nabla \dot{{\bf u}}+\nabla \cdot{\bf q}-H=0,$

(6.1)

where

ρ

is the mass density,

w

is the stored energy per unit mass,

T

is Cauchy stress,

u

is the displacement field,

q

is heat flux, and

H

is the rate of electromagnetic energy absorbed due to Joule-heating (a source term)

$H=a\left(\bf J \cdot \bf E\right),$

(6.2)

where 0 ≤ 

a

 ≤ 1 is an absorption constant. This standard form of Joule-heating is derived in the next chapter. Thus, a material designer must seek ways by which to modify a base material in order to deliver a specified overall conductivity (for example, by employing particulate additives), while simultaneously avoiding overheating.