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

TL;DR

This paper develops a method to derive tight bounds on the time-dependent stress and strain responses of viscoelastic composites, using an analytic approach combined with linear programming to optimize over pole configurations.

Contribution

It applies the analytic method to the time domain for antiplane viscoelasticity, enabling accurate bounds on transient responses based on composite properties.

Findings

Bounds are very accurate and can be tight over the entire time range.

Incorporating volume fractions improves bound tightness at specific times.

Method effectively predicts transient behavior of viscoelastic composites.

Abstract

To derive bounds on the strain and stress response of a two-component composite material with viscoelastic phases, we revisit the so-called analytic method (Bergman 1978), which allows one to approximate the complex effective tensor, function of the ratio of the component shear moduli, as the sum of poles weighted by positive semidefinite residue matrices. The novelty of this investigation lies in the application of such a method, previously applied (Milton 1980; Bergman 1980) to problems involving cyclic loadings in the frequency domain, to derive bounds in the time domain for the antiplane viscoelasticity case. The position of the poles and the residues matrices are the variational parameters of the problem: the aim is to determine such parameters in order to have the minimum (or maximum) response at any given moment in time. All the information about the composite is translated for each fixed pole configuration into constraints on the residues, the so-called sum rules. The linearity of the constraints, along with the observation that the response at a fixed time is linear in the residues, enables one to use the theory of linear programming to reduce the problem to one involving relatively few non-zero residues. Finally, bounds on the response are obtained by numerically optimizing over the pole positions. In the examples studied, the results turn out to be very accurate estimates: if sufficient information about the composite is available, the bounds can be quite tight over the entire range of time, allowing one to predict the transient behavior of the composite. Furthermore, the bounds incorporating the volume fractions (and possibly transverse isotropy) can be extremely tight at certain specific times.