Thermodynamic functions for ageing viscoelasticity: Integral form without internal variables

Viscoelastic materials whose creep and relaxation functions depend on the age at loading are considered. First the material is assumed to follow the solidification theory, which explains ageing of concrete by gradual deposition of layers of newly solidified non-ageing viscoelastic constituent (cement gel ) on the pore walls. It is shown that the well-known classical Staverman and Schwarzls and Mandels formulae for the densities of Helmholtz free energy, free enthalpy and dissipation in a non-ageing viscoelastic material can be generalized to the case of ageing by double Stieltjes integrals over the strain or stress histories. Their integrands contain only the relaxation or compliance functions and their rates, but not the internal variables nor the properties of constituents. The expressions obtained for the free energy and free enthalpy are subsequently shown to possess in general (without restriction to the solidification theory) the property of a potential for the stress or strain, respectively. Consequently, if one formulates three-dimensional expressions for the free energy or free enthalpy on the basis of the solidification theory, one may, conversely, obtain constitutive equations for ageing viscoelasticity that are consistent with continuum thermodynamics. An expression for the dissipated power of an ageing material is also derived. The results should prove useful for approximate solutions, bounds on structural response, and numerical solution algorithms.