Proof of Straughan’s claim on Payne–Song’s and modified Guyer–Krumhansl’s equations

We prove a claim by Brian Straughan who conjectured that Payne–Song’s and modified Guyer–Krumhansl’s equations can be justified in (and derived from) the general model-building framework for the mechanics of complex bodies, so they emerge from the modelling of microstructural effects. The proof is based on taking into account micro-to-macro spatial scaling and a decomposition of the heat flux into a standard Fourier- type component and one measuring microstructural event with associated and balanced actions.