An explicit formula for the minimum free energy in linear viscoelasticity

A general explicit formula for the maximum recoverable work from a given st ate is derived in the frequency domain for full tensorial isothermal linear viscoelastic constitutive equations. A variational approach, developed for the scalar case, is here generalized by virtue of certain factorizability properties of positive-definite matrices. The resultant formula suggests ho w to characterize the state in the sense of Noll in the frequency domain. T he property that the maximum recoverable work represents the minimum free e nergy according to both Graffi's and Coleman-Owen's definitions is used to obtain an explicit formula for the minimum free energy. Detailed expression s are presented for particular types of relaxation function.