Bounds on the effective properties of composites in the time domain
Consider a two-component viscoelastic composite subject to antiplane shear. What is the minimum or maximum stress at each moment of time, when the composite, for instance, is subject to a relaxation experiment (a strain field of the Heaviside type is applied)? By using the analyticity properties of the effective tensor we determine bounds in the time domain for several scenarios in which different pieces of information about the composite are known.
Figure 1. Comparison between the lower and upper bounds on the stress response σ12(t), in case phase 2 is elastic and phase 1 is modeled as a Maxwell viscoelastic material, for the following three cases: no information about the composite is given, the volume fraction of the components is known (f1 = 0.4), and the composite is isotropic with given volume fractions. When information such as the volume fraction of the components or the value of the response at a specific time is considered, the bounds are quite tight over the entire range of time, thus allowing one to predict the transient behavior of the composite. Most noticeably, when the volume fraction is known, the bounds are extremely tight at certain specific times, suggesting the possibility of measuring the response of the composite at such times and, by using the bounds in an inverse fashion, almost exactly determining the volume fraction of the components of the composite.
Figure 2. Comparison between the lower and upper bounds on the stress response σ12(t), in case phase 2 is elastic and phase 1 is modeled as a Maxwell viscoelastic material, for the following three cases: no information about the composite is given, the value of σ12(t) at t = 0 is prescribed, and the value of σ12(t) at t = 0 and the volume fractions are known (f1 = 0.4).
Figure 3. Results similar to those of Figure 1 have also been obtained for the problem of bounding the strain response in time when a Heaviside-type stress field is applied (the so-called creep test). Here we compare the lower and upper bounds on the strain response ε12(t), in case phase 2 is elastic and phase 1 is modeled as a Kelvin-Voigt viscoelastic material in the following three cases: no information about the composite is given, the volume fraction of the components is known (f1 = 0.4), and the composite is isotropic with given volume fractions.