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

Abstract

In order 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 the present 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, such as the knowledge of the volume fractions or the transverse isotropy of the composite, is translated for each fixed pole configuration into (linear) constraints on the residues, the so-called sum rules. Further constraints can be obtained from the knowledge of the response of the composite at specific times (in this paper, for instance, we show how one can include information about the instantaneous and the long-term response of the composite).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: thus measuring the response at such times, and using the bounds in an inverse fashion, gives very tight bounds on the volume fraction of the phases in the composite.

An extremal problem arising in the dynamics of two-phase materials that directly reveals information about the internal geometry

“…The motivation for studying this problem comes from our work [31,33] where we derived microstructure-independent bounds on the viscoelastic response at a given time t of twophase periodic composites (in antiplane shear) with prescribed volume fractions f 1 and f 2 = 1 − f 1 of the phases and with an applied average stress or strain prescribed as a function of time. We found that the bounds were sometimes extremely tight at particular times t = t 0 : see Figure 1.…”

“…The bounds become tighter and tighter as more information on the composite structure is included, so that if color is missing from the figure the outermost pair of bounds are those with no information, the middle pair include just the volume fraction, and the innermost pair include both volume fraction and isotropy. Reproduced from Figure 6.2 in [33].…”

“…However the bounds remain quite wide except near t = 0. Reproduced from Figure 6.5 in [33]. The approach developed in this paper can yield tight bounds with a suitably designed input function as shown in Figure 3.…”

“…The real constant a 0 > 0 and the unknown measure dµ depend on the system. In our viscoelasticity study [33] the connection with Markov functions comes from the fact that the effective shear modulus G * (ω), that relates the average stress to the average strain at frequency ω, as a function of the shear moduli G 1 (ω) and G 2 (ω) of the two phases, has the property that…”

On the Range of Complex Effective Permittivities of Isotropic Two-Phase Composites and Related Problems

“…We remark, in passing, that the analytic method is also useful for bounding the response in the time domain for a given time dependent applied field (that is not at constant frequency) [22]. This approach is more useful for the equivalent antiplane elasticity problem since typical relaxation times are much longer.…”