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To derive bounds on the strain and stress response of a two-componentcomposite material with viscoelastic phases, we revisit the so-called analyticmethod (Bergman 1978), which allows one to approximate the complex effectivetensor, function of the ratio of the component shear moduli, as the sum ofpoles weighted by positive semidefinite residue matrices. The novelty of thisinvestigation lies in the application of such a method, previously applied(Milton 1980; Bergman 1980) to problems involving cyclic loadings in thefrequency domain, to derive bounds in the time domain for the antiplaneviscoelasticity case. The position of the poles and the residues matrices arethe variational parameters of the problem: the aim is to determine suchparameters in order to have the minimum (or maximum) response at any givenmoment in time. All the information about the composite is translated for eachfixed pole configuration into constraints on the residues, the so-called sumrules. The linearity of the constraints, along with the observation that theresponse at a fixed time is linear in the residues, enables one to use thetheory of linear programming to reduce the problem to one involving relativelyfew non-zero residues. Finally, bounds on the response are obtained bynumerically optimizing over the pole positions. In the examples studied, theresults turn out to be very accurate estimates: if sufficient information aboutthe composite is available, the bounds can be quite tight over the entire rangeof time, allowing one to predict the transient behavior of the composite.Furthermore, the bounds incorporating the volume fractions (and possiblytransverse isotropy) can be extremely tight at certain specific times.

Bounds for the response of viscoelastic composites under antiplane loadings in the time domain