Further inequalities for viscoelastic relaxation functions
type: Item
instance of: scholarly article
title: Further inequalities for viscoelastic relaxation functions
zbMATH Open document ID: 0852.73025
DOI: 10.1016/0093-6413(95)00035-P
published in: Mechanics Research Communications
publication date: 1996-02-04T00:00:00Z
review text: 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 \({\mathcal G}\).
Mathematics Subject Classification ID: 74D05
Mathematics Subject Classification ID: 74D10
Mathematics Subject Classification ID: 74A15
zbMATH DE Number: 831865
zbMATH Keywords: constitutive equation
zbMATH Keywords: linear viscoelastic solid
zbMATH Keywords: viscoelastic fluid
zbMATH Keywords: Cauchy stress tensor
zbMATH Keywords: velocity gradient
zbMATH Keywords: second law of thermodynamics
zbMATH Keywords: positive definiteness
zbMATH Keywords: Laplace transforms
author: Angelo Morro
author: Mauro Fabrizio
reviewed by: Gheorghe Gr. Ciobanu
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 831865