Further inequalities for viscoelastic relaxation functions (Q1905523)

The constitutive equation of a linear viscoelastic solid \(T(t)= G_0 E(t)+ \int^\infty_0 G' (\tau) E(t- \tau) d\tau\) and of a viscoelastic fluid \(T(t)= -p(\rho (t)) 1+\int^\infty_0 {\mathcal G} (\tau) D(t- \tau) d\tau\) are considered. Here \(T\in \text{Sym}\) is the Cauchy stress tensor, \(G_0\in \text{Lin(Sym)}\) is the instantaneous elastic modulus, \(G'\in \mathbb{R} \mapsto \text{Lin(Sym)}\) and \({\mathcal G}: \mathbb{R} \mapsto \text{Lin(Sym)}\) are the relaxation tensors, \(\mathbb{R}^+= [0, \infty)\), \(p\) is the pressure, \(\rho\) the density, and \(D\) is the symmetric part of the velocity gradient [see the authors' monograph, Mathematical problems in linear viscoelasticity, SIAM Stud. Appl. Math. 12 (1992; Zbl 0753.73003)]. The authors prove that the second law of thermodynamics implies the positive definiteness of the fourth order tensors \({{DG_L'} \over {du}} (u)\) and \({\mathcal G}_L (u)\) on \(\mathbb{R}^+\), where \(G_L'\) and \({\mathcal G}_L\) are the Laplace transforms of the relaxation tensors \(G'\) and \({\mathcal G}\).