Variational formulations for the linear viscoelastic problem in the time domain

• We use a bilinear form of the convolutive type in the time variable.
• The time domain is decomposed into two equal subintervals, with the consequent decomposition of the constitutive law.
• From the split constitutive law operator, we isolate a symmetric and positive definite sub-operator and we rephrase the constitutive law.
• Five new variational formulations, one of which is of the minimum type, are derived.
• Bounds of the homogenized mechanical properties of viscoelastic composites are obtained by using the minimum variational principle here derived.

Under the assumption of small displacements and strains, we formulate new variational principles for the linear viscoelastic hereditary problem, extending the well-known Hu-Washizu, Hellinger-Reissner, Total Potential Energy, and Complementary Energy principles related to the purely elastic problem. In addition, a new global minimum formulation is derived, giving an energetic interpretation. The new formulations are based on a convolutive bilinear form of the Stieltjes type and on the division of the time domain into two equal parts, with the resulting decomposition of the variables and of the equations governing the problem. In particular, the global minimum principle is achieved by virtue of the positive definiteness of a part of the split constitutive law operator and by means of a partial Legendre transform, and is then used to provide bounds of the overall mechanical properties of viscoelastic composite materials.