Stable fitting of noisy stress relaxation data

Abstract

The mechanical behavior of a viscoelastic material can often be described by a spectrum of Maxwell-Wiechert elements. The inverse problem associated with estimating the parameters of this spectrum from the results of viscoelastic relaxation (ramp-and-hold) test is in general ill-posed and unstable, with estimates highly sensitive to initial conditions and noise. Here, we demonstrate stable estimation of a continuous viscoelastic spectrum from stress relaxation experiments using Tikhonov regularization. We assess the effects of noise and sampling frequency on these estimates, and describe regularization parameter selection. We demonstate the algorithm by estimating the viscoelastic relaxation spectra of soft vinyl samples.Keywords Viscoelastic material characterization · Viscoelastic spectrum · Noise · Ill-posed inverse problems · Tikhonov regularization IntroductionThe mechanical responses of nearly all structures and materials, especially soft materials, depend upon both the rate at which they are deformed, and their history of prior loading [19]. Engineers classify such materials as viscoelastic. Engineering models of viscoelastic behavior describe, on a phenomological level, the storage, transmission, and dissipation of energy within the material. Although typical engineering models of viscoelasticity do not attempt to directly describe the mechanisms governing these processes, they have been successfully used to study causal links between these small-scale processes and the overall mechanical response of the material in a variety of fields, ranging from biology to solid state physics [29].In the body, cells and tissues play both structural and mechanical roles. Studies have revealed mechanical responses consistent with the engineering understanding of viscoelasticity in collagen-rich extracellular matrices (ECMs) [41], single cells [32,53], and subcellular structures such as the nucleus [21]. With the individual building blocks displaying viscoelastic

Effects of Hydrophobic Modifications on the Solution Self-Assembly of P(DMAEMA-co-QDMAEMA)-b-POEGMA Random Diblock Copolymers

“…All obtained by ALV GmbH, Langen (Hessen), Germany. Autocorrelation functions were analyzed by the cumulants method [ 57 ] and the CONTIN algorithm [ 58 ]. Before measurements, dust particles were removed from the solutions by filtration through 0.45 mm hydrophilic PVDF filters.…”