Stable fitting of noisy stress relaxation data
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ORIGINAL PAPER
Stable fitting of noisy stress relaxation data Roger A. Rowe1 · Kenneth M. Pryse1 · Elliot L. Elson2 · Guy M. Genin1 Received: 12 May 2019 / Accepted: 16 July 2019 © Springer Nature Switzerland AG 2019
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
1 Introduction The 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
Guy M. Genin
[email protected] Roger A. Rowe [email protected] Kenneth M. Pryse [email protected] Elliot L. Elson [email protected] 1
NSF Science and Technology Center for Engineering Mechanobiology, Department of Mechanical Engineering and Materials Science, Washington University in St. Louis, St. Louis, MO 63130, USA
2
Department of Biochemistry and Molecular Biophysics, Washington University School of Medicine, St. Louis, MO 60613, USA
R. A. Rowe et al.
behavior, it is of little surprise that mechanical responses characteristic of viscoelastic materials have also been well documented in a variety of bulk tissues and whole organs in all regions of the body, such as arteries [13], brain parenchyma [45], liver tissue [10], the prostate [43], skin [11], breasts [9], articular cartilage [14], ligaments [2, 7], and tendon [17]. Si
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