Comparison and minimum theorems in the time domain for viscoelasticity and other convolutive initial-boundary value problems with applications to random inhomogeneous materials

Abstract

The new variational theorems presented in Huet (J. Mech. Phys. Solids 49 (2001) 675–706) for viscoelasticity problems with applications to heterogeneous materials are supplemented by considering functionals involving the rates of the relaxation and creep kernels and the creep kernel itself. As in Huet (J. Mech. Phys. Solids 49 (2001) 675–706), the results are based on a Continuum Thermodynamics approach, from which extended Clapeyron formulae are derived, leading to the use of auxiliary problems involving data continuations. Functional requirements are discussed with more details. Extended comparison theorems and a new set of minimum theorems are derived. The latter supplement the ones obtained by Huet (Mech. Mater. 31 (1999) 787–829) for other functionals under stronger admissibility conditions. The results are applied to the derivation of a further set of order relationships in the time domain for the overall properties of viscoelastic heterogeneous media supplementing the ones obtained in Huet (J. Mech. Phys. Solids 49 (2001) 675–706) for the boundary-condition effects in bodies smaller than the representative volume and in Huet (Mech. Mater. 31 (1999) 787–829) for the size-effects. This provides the viscoelastic counterpart of the hierarchies first derived by Huet (J. Mech. Phys. Solids 38 (1990) 813–841; Engng Fract. Mech. 58 (5–6) (1997) 459–556) for the elastic case without and with unloaded defects that are recovered as particular cases. A viscoelastic extension of the classical Hill (J. Mech. Phys. Solids 11 (1963) 357–372) modification theorem is also derived for a special class of kernel modifications. The obtained results apply to other kinds of time-dependent physical properties like dielectric and piezoelectric relaxation and thermal or mass diffusion problems, under appropriate choice of the variables respecting the needed positive definiteness requirements provided by the positiveness of the dissipated power and the entropy production rate.