Stability bounds of a delay visco-elastic rheological model with substrate friction

The paper’s linear model yields the characteristic equation f(m)=m^2+γ m e^{-mτ}+((k1+k2)/η) m+(γ k1/η) e^{-mτ} (their Eq. (9)) and proves the sufficient stability condition k1+k2−γη−k1γτ>0 via D-curves and a theorem of Stépán (Proposition 1) . The candidate solution derives the same characteristic equation, reduces to parameters α=(k1+k2)/η, β=(γ k1)/η, excludes imaginary-axis roots using sin x≤x and cos x≥−1, and then invokes continuity of retarded DDE roots from τ=0 to the given τ to conclude stability when α−γ−βτ>0, i.e., k1+k2−γη−k1γτ>0. Thus both arrive at the identical sufficient bound; the paper uses D-curve machinery while the model uses a direct imaginary-root exclusion plus root-continuity. The conclusions match and are correct for this sufficient condition.