_T (_) ≈
E
ρs0cvV
= 3  ̄ qδ
ρs0cvR(1 − _3)(D −  ̄u)
, (43)
where _ = ˆR/R is an energy localization parameter that varies
over the range 0 ≤ _ ≤ 1. For _ = 0, locally dissipated
energy is deposited uniformly throughout the grain resulting
in bulk heating. For _ = 1, dissipated energy is deposited
at the surface having zero volume; thus, this condition is a
singularity in Eq. (43). An estimate for _ based on the thermal
diffusivity of the solid, α, and the spatial duration for
inelastic heating is given by
_ = 1 − 1
R
αδ
D −  ̄u
.