Foams have unique rheological properties that range from solid-like to fluid-like. We study twodimensional non-coarsening foams of different disorder under shear in a Monte Carlo simulation, using a driven large-Q Potts model. Simulations of periodic shear on an ordered foam show several different response regimes. At small strain amplitudes, bubbles deform and recover their shapes elastically, and the macroscopic response is that of a linear elastic cellular material. For increasing strain amplitude, the energy-strain curve starts to exhibit hysteresis before any topological rearrangements occur, indicating a macroscopic viscoelastic response. When the applied strain amplitude exceeds a critical value, the yield strain, topological rearrangements (T1 events) occur, the foam starts to flow, and we observe macroscopic irreversibility. We find that the dynamics of topological rearrangements depend sensitively on the structural disorder. Structural disorder decreases the yield strain; sufficiently high disorder changes the macroscopic response of a foam from a viscoelastic solid to a viscoelastic fluid. This wide-ranging dynamical response and the associated history effects of foams result from avalanche-like T1 events. The spatio-temporal statistics of T1 events T1 do not display long-range correlations for ordered foams or at low shear rates, consistent with experimental observations. As the shear rate or structural disorder increases, the topological events become more correlated and their power spectra change from that of white noise toward 1/f noise. Intriguingly, the power spectra of the total stored energy also exhibit this 1/f trend.

In a continuum description of materials, the stress tensor field σ quantifies the internal forces the neighbouring regions exert on a region of the material.

We present an analysis of the accuracy and information content of three-dimensional reconstructions of the dielectric susceptibility of a sample from noisy, near-field holographic measurements, such as those made in scanning probe microscopy. Holographic measurements are related to the dielectric susceptibility via a linear operator within the accuracy of the first Born approximation. The maximum-likelihood reconstruction of the dielectric susceptibility is expressed as a linear combination of basis functions determined by singular value decomposition of the weighted measurement operator. Maximum a posteriori estimates based on prior information are also discussed. Semianalytical expressions are given for the likely error due to measurement noise in the basis function coefficients, resulting in effective resolution limits in all three dimensions. These results are illustrated by numerical examples.

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