Fractional viscoelastic models for power-law materials

Unveiling the role of hydroxyapatite and hydroxyapatite/silver composite in osteoblast-like cell mineralization: An exploration through their viscoelastic properties

Mechanical properties are becoming fundamental for advancing the comprehension of cellular processes. This study addresses the relationship between viscoelastic properties and the cellular mineralization process. Osteoblast-like cells treated with an osteogenic medium were employed for this purpose. Additionally, the study explores the impact of hydroxyapatite (HA) and hydroxyapatite/silver (HA/Ag) composite on this process. AFM relaxation experiments were conducted to extract viscoelastic parameters using the Fractional Zener (FZ) and Fractional Kelvin (FK) models. Our findings revealed that the main phases of mineralization are associated with alterations in the viscoelastic properties of osteoblast-like cells. Furthermore, HA and HA/Ag treatments significantly influenced changes in the viscoelastic properties of these cells. In particular, the HA/Ag treatment demonstrated a marked enhancement in cell fluidity, suggesting a possible role of silver in accelerating the mineralization process. Moreover, the study underscores the independence observed between fluidity and stiffness, indicating that modifications in one parameter may not necessarily correspond to changes in the other. These findings shed light on the factors involved in the cellular mineralization process and emphasize the importance of using viscoelastic properties to discern the impact of treatments on cells.

Sensitivity of magnetic resonance elastography to extracellular matrix and cell motility in human prostate cancer cell line-derived xenograft models

Prostate cancer (PCa) is a significant health problem in the male population of the Western world. Magnetic resonance elastography (MRE), an emerging medical imaging technique sensitive to mechanical properties of biological tissues, detects PCa based on abnormally high stiffness and viscosity values. Yet, the origin of these changes in tissue properties and how they correlate with histopathological markers and tumor aggressiveness are largely unknown, hindering the use of tumor biomechanical properties for establishing a noninvasive PCa staging system. To infer the contributions of extracellular matrix (ECM) components and cell motility, we investigated fresh tissue specimens from two PCa xenograft mouse models, PC3 and LNCaP, using magnetic resonance elastography (MRE), diffusion-weighted imaging (DWI), quantitative histology, and nuclear shape analysis. Increased tumor stiffness and impaired water diffusion were observed to be associated with collagen and elastin accumulation and decreased cell motility. Overall, LNCaP, while more representative of clinical PCa than PC3, accumulated fewer ECM components, induced less restriction of water diffusion, and exhibited increased cell motility, resulting in overall softer and less viscous properties. Taken together, our results suggest that prostate tumor stiffness increases with ECM accumulation and cell adhesion - characteristics that influence critical biological processes of cancer development. MRE paired with DWI provides a powerful set of imaging markers that can potentially predict prostate tumor development from benign masses to aggressive malignancies in patients. STATEMENT OF SIGNIFICANCE: Xenograft models of human prostate tumor cell lines, allowing correlation of microstructure-sensitive biophysical imaging parameters with quantitative histological methods, can be investigated to identify hallmarks of cancer.

Comprehensive Characterization of the Viscoelastic Properties of Bovine Submaxillary Mucin (BSM) Hydrogels and the Effect of Additives

This study presents a comprehensive characterization of the viscoelastic and structural properties of bovine submaxillary mucin (BSM), which is widely used as a commercial source to conduct mucus-related research. We conducted concentration studies of BSM and examined the effects of various additives, NaCl, CaCl2, MgCl2, lysozyme, and DNA, on its rheological behavior. A notable connection between BSM concentration and viscoelastic properties was observed, particularly under varying ionic conditions. The rheological spectra could be well described by a fractional Kelvin-Voigt model with a minimum of model parameters. A detailed proteomics analysis provided insight into the protein, especially mucin composition within BSM, showing MUC19 as the main component. Cryo-scanning electron microscopy enabled the visualization of the porous BSM network structure. These investigations give us a more profound comprehension of the BSM properties, especially those pertaining to viscoelasticity, and how they are influenced by concentration and environmental conditions, aspects relevant to the field of mucus research.

Characterizing intracellular mechanics via optical tweezers-based microrheology

Intracellular organization is a highly regulated homeostatic state maintained to ensure eukaryotic cells' correct and efficient functioning. Thanks to decades of research, vast knowledge of the proteins involved in intracellular transport and organization has been acquired. However, how these influence and potentially regulate the intracellular mechanical properties of the cell is largely unknown. There is a deep knowledge gap between the understanding of cortical mechanics, which is accessible by a series of experimental tools, and the intracellular situation that has been largely neglected due to the difficulty of performing intracellular mechanics measurements. Recently, tools required for such quantitative and localized analysis of intracellular mechanics have been introduced. Here, we review how these approaches and the resulting viscoelastic models lead the way to a full mechanical description of the cytoplasm, which is instrumental for a quantitative characterization of the intracellular life of cells.

Development of a piezo stack - laser Doppler vibrometer sensing approach for characterizing shear wave dispersion and local viscoelastic property distributions

Laser Doppler vibrometry and wavefield analysis have recently shown great potential for nondestructive evaluation, structural health monitoring, and studying wave physics. However, there are limited studies on these approaches for viscoelastic soft materials, especially, very few studies on the laser Doppler vibrometer (LDV)-based acquisition of time-space wavefields of dispersive shear waves in viscoelastic materials and the analysis of these wavefields for characterizing shear wave dispersion and evaluating local viscoelastic property distributions. Therefore, this research focuses on developing a piezo stack-LDV system and shear wave time-space wavefield analysis methods for enabling the functions of characterizing the shear wave dispersion and the distributions of local viscoelastic material properties. Our system leverages a piezo stack to generate shear waves in viscoelastic materials and an LDV to acquire time-space wavefields. We introduced space-frequency-wavenumber analysis and least square regression-based dispersion comparison to analyze shear wave time-space wavefields and offer functions including extracting shear wave dispersion relations from wavefields and characterizing the spatial distributions of local wavenumbers and viscoelastic properties (e.g., shear elasticity and viscosity). Proof-of-concept experiments were performed using a synthetic gelatin phantom. The results show that our system can successfully generate shear waves and acquire time-space wavefields. They also prove that our wavefield analysis methods can reveal the shear wave dispersion relation and show the spatial distributions of local wavenumbers and viscoelastic properties. We expect this research to benefit engineering and biomedical research communities and inspire researchers interested in developing shear wave-based technologies for characterizing viscoelastic materials.

Filled Elastomers: Mechanistic and Physics-Driven Modeling and Applications as Smart Materials

Elastomers are made of chain-like molecules to form networks that can sustain large deformation. Rubbers are thermosetting elastomers that are obtained from irreversible curing reactions. Curing reactions create permanent bonds between the molecular chains. On the other hand, thermoplastic elastomers do not need curing reactions. Incorporation of appropriated filler particles, as has been practiced for decades, can significantly enhance mechanical properties of elastomers. However, there are fundamental questions about polymer matrix composites (PMCs) that still elude complete understanding. This is because the macroscopic properties of PMCs depend not only on the overall volume fraction (ϕ) of the filler particles, but also on their spatial distribution (i.e., primary, secondary, and tertiary structure). This work aims at reviewing how the mechanical properties of PMCs are related to the microstructure of filler particles and to the interaction between filler particles and polymer matrices. Overall, soft rubbery matrices dictate the elasticity/hyperelasticity of the PMCs while the reinforcement involves polymer-particle interactions that can significantly influence the mechanical properties of the polymer matrix interface. For ϕ values higher than a threshold, percolation of the filler particles can lead to significant reinforcement. While viscoelastic behavior may be attributed to the soft rubbery component, inelastic behaviors like the Mullins and Payne effects are highly correlated to the microstructures of the polymer matrix and the filler particles, as well as that of the polymer-particle interface. Additionally, the incorporation of specific filler particles within intelligently designed polymer systems has been shown to yield a variety of functional and responsive materials, commonly termed smart materials. We review three types of smart PMCs, i.e., magnetoelastic (M-), shape-memory (SM-), and self-healing (SH-) PMCs, and discuss the constitutive models for these smart materials.

Nanoscopic Interfacial Hydrogel Viscoelasticity Revealed from Comparison of Macroscopic and Microscopic Rheology

Deviations between macrorheological and particle-based microrheological measurements are often considered to be a nuisance and neglected. We study aqueous poly(ethylene oxide) (PEO) hydrogels for varying PEO concentrations and chain lengths that contain microscopic tracer particles and show that these deviations reveal the nanoscopic viscoelastic properties of the particle-hydrogel interface. Based on the transient Stokes equation, we first demonstrate that the deviations are not due to finite particle radius, compressibility, or surface-slip effects. Small-angle neutron scattering rules out hydrogel heterogeneities. Instead, we show that a generalized Stokes-Einstein relation, accounting for an interfacial shell around tracers with viscoelastic properties that deviate from bulk, consistently explains our macrorheological and microrheological measurements. The extracted shell diameter is comparable to the PEO end-to-end distance, indicating the importance of dangling chain ends. Our methodology reveals the nanoscopic interfacial rheology of hydrogels and is applicable to different kinds of viscoelastic fluids and particles.

Active viscoelastic models for cell and tissue mechanics

Living cells are out of equilibrium active materials. Cell-generated forces are transmitted across the cytoskeleton network and to the extracellular environment. These active force interactions shape cellular mechanical behaviour, trigger mechano-sensing, regulate cell adaptation to the microenvironment and can affect disease outcomes. In recent years, the mechanobiology community has witnessed the emergence of many experimental and theoretical approaches to study cells as mechanically active materials. In this review, we highlight recent advancements in incorporating active characteristics of cellular behaviour at different length scales into classic viscoelastic models by either adding an active tension-generating element or adjusting the resting length of an elastic element in the model. Summarizing the two groups of approaches, we will review the formulation and application of these models to understand cellular adaptation mechanisms in response to various types of mechanical stimuli, such as the effect of extracellular matrix properties and external loadings or deformations.

Effect of non-linear strain stiffening in eDAH and unjamming

In cell clusters, the prominent factors at play encompass contractility-based enhanced tissue surface tension and cell unjamming transition. The former effect pertains to the boundary effect, while the latter constitutes a bulk effect. Both effects share outcomes of inducing significant elongation in cells. This elongation is so substantial that it surpasses the limits of linear elasticity, thereby giving rise to additional effects. To investigate these effects, we employ atomic force microscopy (AFM) to analyze how the mechanical properties of individual cells change under such considerable elongation. Our selection of cell lines includes MCF-10A, chosen for its pronounced demonstration of the extended differential adhesion hypothesis (eDAH), and MDA-MB-436, selected due to its manifestation of cell unjamming behavior. In the AFM analyses, we observe a common trend in both cases: as elongation increases, both cell lines exhibit strain stiffening. Notably, this effect is more prominent in MCF-10A compared to MDA-MB-436. Subsequently, we employ AFM on a dynamic range of 1-200 Hz to probe the mechanical characteristics of cell spheroids, focusing on both surface and bulk mechanics. Our findings align with the results from single cell investigations. Specifically, MCF-10A cells, characterized by strong contractile tissue tension, exhibit the greatest stiffness on their surface. Conversely, MDA-MB-436 cells, which experience significant elongation, showcase their highest stiffness within the bulk region. Consequently, the concept of single cell strain stiffening emerges as a crucial element in understanding the mechanics of multicellular spheroids (MCSs), even in the case of MDA-MB-436 cells, which are comparatively softer in nature.

Acid-Induced Gelation of Carboxymethylcellulose Solutions

The present work offers a comprehensive description of the acid-induced gelation of carboxymethylcellulose (CMC), a water-soluble derivative of cellulose broadly used in numerous applications ranging from food packaging to biomedical engineering. Linear viscoelastic properties measured at various pH and CMC contents allow us to build a sol-gel phase diagram and show that CMC gels exhibit broad power-law viscoelastic spectra that can be rescaled onto a master curve following a time-composition superposition principle. These results demonstrate the microstructural self-similarity of CMC gels and inspire a mean-field model based on hydrophobic interchain association that accounts for the sol-gel boundary over the entire range of CMC content under study. Neutron scattering experiments further confirm this picture and suggest that CMC gels comprise a fibrous network cross-linked by aggregates. Finally, low-field NMR measurements offer an original signature of acid-induced gelation from a solvent perspective. Altogether, these results open avenues for the precise manipulation and control of CMC-based hydrogels.

Rethinking the relationships between gel like structure and sludge dewaterability based on a binary gel like structure model: Implications for the online sensing of dewaterability

The digital transformation of sludge treatment processes requires online sensing of dewaterability. This topic has been attempted for many years based on macroscopic shear rheology. However, the relationship between rheological behavior and dewaterability remains noncommittal, and the reason is unclear. Herein, a binary gel-like structure model was proposed including the interactions network at the supra-flocs level and the gel-like structure at the flocs level. Multiple advanced techniques including optical tweezers were employed to precisely understand the binary gel-like structure and to classify the correlation mechanism between this gel-like structure, rheological behavior, and dewaterability. The analysis of sludge from eight wastewater treatment plants showed the binary gel-like structures at both supra-flocs and flocs levels have significant relationships with sludge dewaterability (p < 0.05). Further deconstruction of the sludge viscoelastic behavior illustrated that the gel-like structure at the supra-flocs level dominates the rheological behavior of sludge. Moreover, the direct description of the binary gel-like structure in four typical sludge treatment processes highlighted the importance of the flocs level's structure in determining the dewaterability. Overall, this study revealed that shear rheology may prefer to stress the interactions network at the supra-flocs level but mask the flocs level's structure, although the latter is important. This observation may provide a general guideline for the design of robust sensors for dewaterability.

Nanorheology and Nanoindentation Revealed a Softening and an Increased Viscous Fluidity of Adherent Mammalian Cells upon Increasing the Frequency

The nanomechanical response of a cell depends on the frequency at which the cell is probed. The components of the cell that contribute to this property and their interplay are not well understood. Here, two force microscopy methods are integrated to characterize the frequency and/or the velocity-dependent properties of living cells. It is shown on HeLa and fibroblasts, that cells soften and fluidize upon increasing the frequency or the velocity of the deformation. This property was independent of the type and values (25 or 1000 nm) of the deformation. At low frequencies (2-10 Hz) or velocities (1-10 µm s-1 ), the response is dominated by the mechanical properties of the cell surface. At higher frequencies (>10 Hz) or velocities (>10 µm s-1 ), the response is dominated by the hydrodynamic drag of the cytosol. Softening and fluidization does not seem to involve any structural remodeling. It reflects a redistribution of the applied stress between the solid and liquid-like elements of the cell as the frequency or the velocity is changed. The data indicates that the quasistatic mechanical properties of a cell featuring a cytoskeleton pathology might be mimicked by the response of a non-pathological cell which is probed at a high frequency.

Modeling the effects of hydration on viscoelastic properties of nucleus pulposus tissue in shear using the fractional Zener model

Nucleus pulposus (NP) tissue in the intervertebral disc (IVD) is a viscoelastic material exhibiting both solid- and fluid-like mechanical behaviors. Advances in viscoelastic models incorporating fractional calculus, such as the Fractional Zener (FZ) model, have potential to describe viscoelastic behaviors. The objectives of this study were to determine whether the FZ model can accurately describe the shear viscoelastic properties of NP tissue and determine if the fractional order (α) is related to tissue hydration. 30 caudal IVDs underwent equilibrium dialysis in 5% or 25% polyethylene glycol solutions to alter tissue hydration. Excised NP tissue underwent stress relaxation testing in shear and unconfined compression. Stress relaxation data was fitted to the FZ model to obtain viscoelastic properties. In both loading modes, the initial modulus was greater for the less hydrated 25% equilibrated samples compared to 5% with no change in the equilibrium modulus. Samples with lower water content (25% samples) had shorter relaxation times in shear and longer time constants in compression, highlighting the different interactions between the fluid and solid matrix in loading modes. Samples with lower water content had α values closer to 0, indicating that less hydrated samples behaved more solid-like on the viscoelastic spectrum. Tissue hydration correlated with α values for 25% samples in shear. This study demonstrates that the FZ model may be used to describe IVD tissue behavior under both loading modes; however, the greatest utility of the FZ model is in describing flow-independent shear behaviors, and α may inform tissue hydration in shear.

Critical-Like Gelation Dynamics in Cellulose Nanocrystal Suspensions

We use time-resolved mechanical spectroscopy to offer a detailed picture of the gelation dynamics of cellulose nanocrystal (CNC) suspensions following shear cessation in the presence of salt. CNCs are charged, rodlike colloids that self-assemble into various phases, including physical gels serving as soft precursors for biosourced composites. Here, a series of linear viscoelastic spectra acquired across the sol-gel transition of CNC suspensions are rescaled onto two master curves that correspond to a viscoelastic liquid state prior to gelation and to a soft solid state after gelation. These two states are separated by a critical gel point, where all rescaling parameters diverge in an asymmetric fashion yet with exponents that obey hyperscaling relations consistent with previous works on isotropic colloids and polymer gels. Upon varying the salt content, we further show that these critical-like dynamics result in both time-connectivity and time-concentration superposition principles.

A mechanochemical model recapitulates distinct vertebrate gastrulation modes

During vertebrate gastrulation, an embryo transforms from a layer of epithelial cells into a multilayered gastrula. This process requires the coordinated movements of hundreds to tens of thousands of cells, depending on the organism. In the chick embryo, patterns of actomyosin cables spanning several cells drive coordinated tissue flows. Here, we derive a minimal theoretical framework that couples actomyosin activity to global tissue flows. Our model predicts the onset and development of gastrulation flows in normal and experimentally perturbed chick embryos, mimicking different gastrulation modes as an active stress instability. Varying initial conditions and a parameter associated with active cell ingression, our model recapitulates distinct vertebrate gastrulation morphologies, consistent with recently published experiments in the chick embryo. Altogether, our results show how changes in the patterning of critical cell behaviors associated with different force-generating mechanisms contribute to distinct vertebrate gastrulation modes via a self-organizing mechanochemical process.

How to Make the Stress Relaxation Experiment for Polymers More Informative

Different viscoelastic models and characteristics are commonly used to describe, analyze, compare and improve the mechanical properties of polymers. A time-dependent linear relaxation modulus next to frequency-domain storage and loss moduli are the basic rheological material functions of polymers. The exponential Maxwell model and the exponential stretched Kohlrausch-Williams-Watts model are, probably, the most known linear rheological models of polymers. There are different identification methods for such models, some of which are dedicated to specific models, while others are general in nature. However, the identification result, i.e., the best model, always depends on the specific experimental data on the basis of which it was determined. When the rheological stress relaxation test is performed, the data are composed of the sampling instants used in the test and on the measurements of the relaxation modulus of the real material. To build a relaxation modulus model that does not depend on sampling instants is a fundamental concern. The problem of weighted least-squares approximation of the real relaxation modulus is discussed when only the noise-corrupted time-measurements of the relaxation modulus are accessible for identification. A wide class of models, that are continuous, differentiable and Lipschitz with respect to parameters, is considered for the relaxation modulus approximation. The main results concern the models that are selected asymptotically as the number of measurements tends to infinity. It is shown that even when the true relaxation modulus description is completely unknown, the approximate optimal model parameters can be derived from the measurement data that are obtained for sampling instants that are selected randomly due to the appropriate randomization introduced whenever certain conditions regarding the adopted class of models are satisfied. It is shown that the most commonly used stress relaxation models, the Maxwell and Kohlrausch-Williams-Watts models, satisfy these conditions. Since the practical problems of the identification of relaxation modulus models are usually ill posed, Tikhonov regularization is applied to guarantee the stability of the regularized solutions. The approximate optimal model is a strongly consistent estimate of the regularized model that is optimal in the sense of the deterministic integral weighted square error. An identification algorithm leading to the best regularized model is presented. The stochastic-type convergence analysis is conducted for noise-corrupted relaxation modulus measurements, and the exponential convergence rate is proved. Numerical studies for different models of the relaxation modulus used in the polymer rheology are presented for the material described by a bimodal Gauss-like relaxation spectrum. Numerical studies have shown that if appropriate randomization is introduced in the selection of sampling instants, then optimal regularized models of the relaxation modulus being asymptotically independent of these time instants can be recovered from the stress relaxation experiment data. The robustness of the identification algorithm to measurement noises was demonstrated both by analytical and numerical analyses.

A multiscale theory for spreading and migration of adhesion-reinforced mesenchymal cells

We present a chemomechanical whole-cell theory for the spreading and migration dynamics of mesenchymal cells that can actively reinforce their adhesion to an underlying viscoelastic substrate as a function of its stiffness. Our multiscale model couples the adhesion reinforcement effect at the subcellular scale with the nonlinear mechanics of the nucleus-cytoskeletal network complex at the cellular scale to explain the concurrent monotonic area-stiffness and non-monotonic speed-stiffness relationships observed in experiments: we consider that large cell spreading on stiff substrates flattens the nucleus, increasing the viscous drag force on it. The resulting force balance dictates a reduction in the migration speed on stiff substrates. We also reproduce the experimental influence of the substrate viscosity on the cell spreading area and migration speed by elucidating how the viscosity may either maintain adhesion reinforcement or prevent it depending on the substrate stiffness. Additionally, our model captures the experimental directed migration behaviour of the adhesion-reinforced cells along a stiffness gradient, known as durotaxis, as well as up or down a viscosity gradient (viscotaxis or anti-viscotaxis), the cell moving towards an optimal viscosity in either case. Overall, our theory explains the intertwined mechanics of the cell spreading, migration speed and direction in the presence of the molecular adhesion reinforcement mechanism.

Heterogeneous anomalous transport in cellular and molecular biology

It is well established that a wide variety of phenomena in cellular and molecular biology involve anomalous transport e.g. the statistics for the motility of cells and molecules are fractional and do not conform to the archetypes of simple diffusion or ballistic transport. Recent research demonstrates that anomalous transport is in many cases heterogeneous in both time and space. Thus single anomalous exponents and single generalised diffusion coefficients are unable to satisfactorily describe many crucial phenomena in cellular and molecular biology. We consider advances in the field ofheterogeneous anomalous transport(HAT) highlighting: experimental techniques (single molecule methods, microscopy, image analysis, fluorescence correlation spectroscopy, inelastic neutron scattering, and nuclear magnetic resonance), theoretical tools for data analysis (robust statistical methods such as first passage probabilities, survival analysis, different varieties of mean square displacements, etc), analytic theory and generative theoretical models based on simulations. Special emphasis is made on high throughput analysis techniques based on machine learning and neural networks. Furthermore, we consider anomalous transport in the context of microrheology and the heterogeneous viscoelasticity of complex fluids. HAT in the wavefronts of reaction-diffusion systems is also considered since it plays an important role in morphogenesis and signalling. In addition, we present specific examples from cellular biology including embryonic cells, leucocytes, cancer cells, bacterial cells, bacterial biofilms, and eukaryotic microorganisms. Case studies from molecular biology include DNA, membranes, endosomal transport, endoplasmic reticula, mucins, globular proteins, and amyloids.

A MEC-2/stomatin condensate liquid-to-solid phase transition controls neuronal mechanotransduction during touch sensing

A growing body of work suggests that the material properties of biomolecular condensates ensuing from liquid-liquid phase separation change with time. How this aging process is controlled and whether the condensates with distinct material properties can have different biological functions is currently unknown. Using Caenorhabditis elegans as a model, we show that MEC-2/stomatin undergoes a rigidity phase transition from fluid-like to solid-like condensates that facilitate transport and mechanotransduction, respectively. This switch is triggered by the interaction between the SH3 domain of UNC-89 (titin/obscurin) and MEC-2. We suggest that this rigidity phase transition has a physiological role in frequency-dependent force transmission in mechanosensitive neurons during body wall touch. Our data demonstrate a function for the liquid and solid phases of MEC-2/stomatin condensates in facilitating transport or mechanotransduction, and a previously unidentified role for titin homologues in neurons.

A viscoelastic constitutive model for human femoropopliteal arteries

High failure rates present challenges for surgical and interventional therapies for peripheral artery disease of the femoropopliteal artery (FPA). The FPA's demanding biomechanical environment necessitates complex interactions with repair devices and materials. While a comprehensive understanding of the FPA's mechanical characteristics could improve medical treatments, the viscoelastic properties of these muscular arteries remain poorly understood, and the constitutive model describing their time-dependent behavior is absent. We introduce a new viscoelastic constitutive model for the human FPA grounded in its microstructural composition. The model is capable of detailing the contributions of each intramural component to the overall viscoelastic response. Our model was developed utilizing fractional viscoelasticity and tested using biaxial experimental data with hysteresis and relaxation collected from 10 healthy human subjects aged 57 to 65 and further optimized for high throughput and automation. The model accurately described the experimental data, capturing significant nonlinearity and hysteresis that were particularly pronounced circumferentially, and tracked the contribution of passive smooth muscle cells to viscoelasticity that was twice that of the collagen fibers. The high-throughput parameter estimation procedure we developed included a specialized objective function and modifications to enhance convergence for the common exponential-type fiber laws, facilitating computational implementation. Our new model delineates the time-dependent behavior of human FPAs, which will improve the fidelity of computational simulations investigating device-artery interactions and contribute to their greater physical accuracy. Moreover, it serves as a useful tool to investigate the contribution of arterial constituents to overall tissue viscoelasticity, thereby expanding our knowledge of arterial mechanophysiology. STATEMENT OF SIGNIFICANCE: The demanding biomechanical environment of the femoropopliteal artery (FPA) necessitates complex interactions with repair devices and materials, but the viscoelastic properties of these muscular arteries remain poorly understood with the constitutive model describing their time-dependent behavior being absent. We hereby introduce the first viscoelastic constitutive model for the human FPA grounded in its microstructures. This model was tested using biaxial mechanical data collected from 10 healthy human subjects between the ages of 57 to 65. It can detail the contributions of each intramural component to the overall viscoelastic response, showing that the contribution of passive smooth muscle cells to viscoelasticity is twice that of collagen fibers. The usefulness of this model as tool to better understand arterial mechanophysiology was demonstrated.

Disrupted Stiffness Ratio Alters Nuclear Mechanosensing

The ability of endothelial cells to sense and respond to dynamic changes in blood flow is critical for vascular homeostasis and cardiovascular health. The mechanical and geometric properties of the nuclear and cytoplasmic compartments affect mechanotransduction. We hypothesized that alterations to these parameters have resulting mechanosensory consequences. Using atomic force microscopy and mathematical modeling, we assessed how the nuclear and cytoplasmic compartment stiffnesses modulate shear stress transfer to the nucleus within aging endothelial cells. Our computational studies revealed that the critical parameter controlling shear transfer is not the individual mechanics of these compartments, but the stiffness ratio between them. Replicatively aged cells had a reduced stiffness ratio, attenuating shear transfer, while the ratio was not altered in a genetic model of accelerated aging. We provide a theoretical framework suggesting that dysregulation of the shear stress response can be uniquely imparted by relative mechanical changes in subcellular compartments.

Sound trapping and waveguiding in locally resonant viscoelastic phononic crystals

We describe the trapping and absorption of audible sound in centimeter-scale claddings of two-dimensional, locally resonant phononic crystals. In a square lattice of local resonators consisting of steel cores and cellulose shells, embedded in a viscous foam, dual acoustic-range band gaps extending from about 200 to [Formula: see text] are achieved. The spectral range consists of a low-frequency, local resonance gap, separated from a higher frequency Bragg resonance gap, by narrow bands of slow-sound modes. We demonstrate that thin claddings of such phononic crystal, of only three unit cells in thickness, can effectively prevent sound transmission, by a combination of reflection and absorption, over much of the audible spectrum. Moreover, frequency-selective sound transmission can be enabled by engineering waveguide channels that transmit sound through the local resonance gap, the Bragg gap, or both. This offers a path to sound-sculpting claddings that can surround a noise-generating source. The viscoelastic foam in our cladding is treated using a fractional Voigt model, capable of describing experimentally observed responses.

On Applicability of the Relaxation Spectrum of Fractional Maxwell Model to Description of Unimodal Relaxation Spectra of Polymers

The relaxation time and frequency spectra are vital for constitutive models and for insight into the viscoelastic properties of polymers, since, from the spectra, other material functions used to describe rheological properties of various polymers can be uniquely determined. In recent decades the non-integer order differential equations have attracted interest in the description of time-dependent processes concerning relaxation phenomena. The fractional Maxwell model (FMM) is probably the most known rheological model of non-integer order. However, the FMM spectrum has not yet been studied and used to describe rheological materials. Therefore, the goal of the present paper was to study the applicability of the relaxation spectrum of FMM to the description of the relaxation spectra of polymers. Based on the known integral representation of the Mittag-Leffler two-parameter function, analytical formulas describing relaxation time and frequency spectra of FMM model were derived. Monotonicity of the spectra was analyzed and asymptotic properties were established. Relaxation frequency spectrum grows for large frequencies with a positive power law, while the relaxation time spectrum decays for large times with a negative power of time. Necessary and sufficient conditions for the existence of the local extrema of the relaxation spectra were derived in the form of two trigonometric inequalities. A simple procedure for checking the existence or absence of the spectra extrema was developed. Direct analytical formulas for the local extrema, minima, and maxima are given in terms of model fractional and viscoelastic parameters. The fractional model parameters, non-integer orders of the stress and strain derivatives of FMM uniquely determine the existence of the spectrum extrema. However, the viscoelastic parameters of the FMM, elastic modulus, and relaxation time affect the maxima and minima of the relaxation spectra and the values of their local peaks. The influence of model parameters on their local extrema was examined. Next, the applicability of the continuous-discrete spectrum of FMM to describe Baumgaertel, Schausberger and Winter (BSW) and unimodal Gauss-like relaxation spectra, commonly used to describe rheological properties of various polymers, was examined. Numerical experiments have shown that by respective choice of the FMM parameters, in particular by respective choice of the orders of fractional derivatives of the stress and strain, a good fit for the relaxation modulus experiment data was obtained for polymers characterized both by BSW and Gauss-like relaxation spectra. As a result, a good approximation of the real spectra was reached. Thus, the viscoelastic relaxation spectrum of FMM, due to the availability of the two extra degrees of freedom (non-integer orders of the stress and strain derivatives), provides deep insights into the complex behavior of polymers and can be applied for a wide class of polymers with unimodal relaxation spectra.

Poloxamer-based nanogels as delivery systems: how structural requirements can drive their biological performance?

Poloxamers or Pluronics®-based nanogels are one of the most used matrices for developing delivery systems. Due to their thermoresponsive and flexible mechanical properties, they allowed the incorporation of several molecules including drugs, biomacromolecules, lipid-derivatives, polymers, and metallic, polymeric, or lipid nanocarriers. The thermogelling mechanism is driven by micelles formation and their self-assembly as phase organizations (lamellar, hexagonal, cubic) in response to microenvironmental conditions such as temperature, osmolarity, and additives incorporated. Then, different biophysical techniques have been used for investigating those structural transitions from the mechanisms to the preferential component's orientation and organization. Since the design of PL-based pharmaceutical formulations is driven by the choice of the polymer type, considering its physico-chemical properties, it is also relevant to highlight that factors inherent to the polymeric matrix can be strongly influenced by the presence of additives and how they are able to determine the nanogels biopharmaceuticals properties such as bioadhesion, drug loading, surface interaction behavior, dissolution, and release rate control. In this review, we discuss the general applicability of three of the main biophysical techniques used to characterize those systems, scattering techniques (small-angle X-ray and neutron scattering), rheology and Fourier transform infrared absorption spectroscopy (FTIR), connecting their supramolecular structure and insights for formulating effective therapeutic delivery systems.

Supplementary Information

The online version contains supplementary material available at 10.1007/s12551-023-01093-2.

Aftershocks and Fluctuating Diffusivity

The Omori-Utsu law shows the temporal power-law-like decrease of the frequency of earthquake aftershocks and, interestingly, is found in a variety of complex systems/phenomena exhibiting catastrophes. Now, it may be interpreted as a characteristic response of such systems to large events. Here, hierarchical dynamics with the fast and slow degrees of freedom is studied on the basis of the Fokker-Planck theory for the load-state distribution to formulate the law as a relaxation process, in which diffusion coefficient in the space of the load state is treated as a fluctuating slow variable. The evolution equation reduced from the full Fokker-Planck equation and its Green's function are analyzed for the subdynamics governing the load state as the fast degree of freedom. It is shown that the subsystem has the temporal translational invariance in the logarithmic time, not in the conventional time, and consequently the aging phenomenon appears.

Compressible viscoelasticity of cell membranes determined by gigahertz-frequency acoustic vibrations

Membrane viscosity is an important property of cell biology, which determines cellular function, development and disease progression. Various experimental and computational methods have been developed to investigate the mechanics of cells. However, there have been no experimental measurements of the membrane viscosity at high-frequencies in live cells. High frequency measurements are important because they can probe viscoelastic effects. Here, we investigate the membrane viscosity at gigahertz-frequencies through the damping of the acoustic vibrations of gold nanoplates. The experiments are modeled using a continuum mechanics theory which reveals that the membranes display viscoelasticity, with an estimated relaxation time of ca. 5.7 + 2.4 / - 2.7 ps. We further demonstrate that membrane viscoelasticity can be used to differentiate a cancerous cell line (the human glioblastoma cells LN-18) from a normal cell line (the mouse brain microvascular endothelial cells bEnd.3). The viscosity of cancerous cells LN-18 is lower than that of healthy cells bEnd.3 by a factor of three. The results indicate promising applications of characterizing membrane viscoelasticity at gigahertz-frequency in cell diagnosis.

Diversity of viscoelastic properties of an engineered muscle-inspired protein hydrogel

Folded protein hydrogels are prime candidates as tuneable biomaterials but it is unclear to what extent their mechanical properties have mesoscopic, as opposed to molecular origins. To address this, we probe hydrogels inspired by the muscle protein titin and engineered to the polyprotein I27₅, using a multimodal rheology approach. Across multiple protocols, the hydrogels consistently exhibit power-law viscoelasticity in the linear viscoelastic regime with an exponent β = 0.03, suggesting a dense fractal meso-structure, with predicted fractal dimension d_f = 2.48. In the nonlinear viscoelastic regime, the hydrogel undergoes stiffening and energy dissipation, indicating simultaneous alignment and unfolding of the folded proteins on the nanoscale. Remarkably, this behaviour is highly reversible, as the value of β, d_f and the viscoelastic moduli return to their equilibrium value, even after multiple cycles of deformation. This highlights a previously unrevealed diversity of viscoelastic properties that originate on both at the nanoscale and the mesoscopic scale, providing powerful opportunities for engineering novel biomaterials.

Viscoelastic Properties of Polymeric Microneedles Determined by Micromanipulation Measurements and Mathematical Modelling

Microneedles, including dissolvable ones made from biocompatible and biodegradable materials, have been widely studied and can potentially be used for transdermal drug delivery, disease diagnosis (sampling), skin care, etc. Characterizing their mechanical properties is essential, as being mechanically strong enough to pierce the skin barrier is one of the most fundamental and crucial requirements for them. The micromanipulation technique was based on compressing single microparticles between two flat surfaces to obtain force and displacement data simultaneously. Two mathematical models had already been developed to calculate the rupture stress and apparent Young's modulus, which can identify variations of these parameters in single microneedles within a microneedle patch. In this study, a new model has been developed to determine the viscoelasticity of single microneedles made of hyaluronic acid (HA) with a molecular weight of 300 kDa loaded with lidocaine by using the micromanipulation technique to gather experimental data. The modelling results from the micromanipulation measurements suggest that the microneedles were viscoelastic and their mechanical behaviour was strain-rate dependent, which implies that the penetration efficiency of viscoelastic microneedles can be improved by increasing their piercing speed into the skin.

Recombinant Peptide Production Softens Escherichia coli Cells and Increases Their Size during C-Limited Fed-Batch Cultivation

Stress-associated changes in the mechanical properties at the single-cell level of Escherichia coli (E. coli) cultures in bioreactors are still poorly investigated. In our study, we compared peptide-producing and non-producing BL21(DE3) cells in a fed-batch cultivation with tightly controlled process parameters. The cell growth, peptide content, and cell lysis were analysed, and changes in the mechanical properties were investigated using atomic force microscopy. Recombinant-tagged somatostatin-28 was expressed as soluble up to 197 ± 11 mg g-1. The length of both cultivated strains increased throughout the cultivation by up to 17.6%, with nearly constant diameters. The peptide-producing cells were significantly softer than the non-producers throughout the cultivation, and respective Young's moduli decreased by up to 57% over time. A minimum Young's modulus of 1.6 MPa was observed after 23 h of the fed-batch. Furthermore, an analysis of the viscoelastic properties revealed that peptide-producing BL21(DE3) appeared more fluid-like and softer than the non-producing reference. For the first time, we provide evidence that the physical properties (i.e., the mechanical properties) on the single-cell level are significantly influenced by the metabolic burden imposed by the recombinant peptide expression and C-limitation in bioreactors.

Optimizing the accuracy of viscoelastic characterization with AFM force-distance experiments in the time and frequency domains

Atomic Force Microscopy (AFM) force-distance (FD) experiments have emerged as an attractive alternative to traditional micro-rheology measurement techniques owing to their versatility of use in materials of a wide range of mechanical properties. Here, we show that the range of time dependent behaviour which can reliably be resolved from the typical method of FD inversion (fitting constitutive FD relations to FD data) is inherently restricted by the experimental parameters: sampling frequency, experiment length, and strain rate. Specifically, we demonstrate that violating these restrictions can result in errors in the values of the parameters of the complex modulus. In the case of complex materials, such as cells, whose behaviour is not specifically understood a priori, the physical sensibility of these parameters cannot be assessed and may lead to falsely attributing a physical phenomenon to an artifact of the violation of these restrictions. We use arguments from information theory to understand the nature of these inconsistencies as well as devise limits on the range of mechanical parameters which can be reliably obtained from FD experiments. The results further demonstrate that the nature of these restrictions depends on the domain (time or frequency) used in the inversion process, with the time domain being far more restrictive than the frequency domain. Finally, we demonstrate how to use these restrictions to better design FD experiments to target specific timescales of a material's behaviour through our analysis of a polydimethylsiloxane (PDMS) polymer sample.

Sounding a New Era in Biomechanics with Acoustic Force Spectroscopy

The acoustic force spectroscopy (AFS) tool was recently introduced as a novel tool for probing mechanical properties of biomolecules, expanding the application of sound waves to high-throughput quantification of the mechanical properties of single cells. By using controlled acoustic forces in the piconewton to nanonewton range, tens to hundreds of cells functionalized by attached microspheres can simultaneously be stretched and tracked in real-time with sub millisecond time response. Since its first application, several studies have demonstrated the potential and versatility of the AFS for high-throughput measurements of force-induced molecular mechanisms, revealing insight into cellular biomechanics and mechanobiology at the molecular level. In this chapter, we describe the operation of the AFS starting with the underlying physical principles, followed by a run-down of experimental considerations, and finally leading to applications in molecular and cellular biology.

Predictive Modeling of Soft Stretchable Nanocomposites Using Recurrent Neural Networks

Human skin is characterized by rough, elastic, and uneven features that are difficult to recreate using conventional manufacturing technologies and rigid materials. The use of soft materials is a promising alternative to produce devices that mimic the tactile capabilities of biological tissues. Although previous studies have revealed the potential of fillers to modify the properties of composite materials, there is still a gap in modeling the conductivity and mechanical properties of these types of materials. While traditional Finite Element approximations can be used, these methodologies tend to be highly demanding of time and processing power. Instead of this approach, a data-driven learning-based approximation strategy can be used to generate prediction models via neural networks. This paper explores the fabrication of flexible nanocomposites using polydimethylsiloxane (PDMS) with different single-walled carbon nanotubes (SWCNTs) loadings (0.5, 1, and 1.5 wt.%). Simple Recurrent Neural Networks (SRNN), Long Short-Term Memory (LSTM), and Gated Recurrent Units (GRU) models were formulated, trained, and tested to obtain the predictive sequence data of out-of-plane quasistatic mechanical tests. Finally, the model learned is applied to a dynamic system using the Kelvin-Voight model and the phenomenon known as the bouncing ball. The best predictive results were achieved using a nonlinear activation function in the SRNN model implementing two units and 4000 epochs. These results suggest the feasibility of a hybrid approach of analogy-based learning and data-driven learning for the design and computational analysis of soft and stretchable nanocomposite materials.

A GENERAL RETURN-MAPPING FRAMEWORK FOR FRACTIONAL VISCO-ELASTO-PLASTICITY

We develop a fractional return-mapping framework for power-law visco-elasto-plasticity. In our approach, the fractional viscoelasticity is accounted through canonical combinations of Scott-Blair elements to construct a series of well-known fractional linear viscoelastic models, such as Kelvin-Voigt, Maxwell, Kelvin-Zener and Poynting-Thomson. We also consider a fractional quasi-linear version of Fung's model to account for stress/strain nonlinearity. The fractional viscoelastic models are combined with a fractional visco-plastic device, coupled with fractional viscoelastic models involving serial combinations of Scott-Blair elements. We then develop a general return-mapping procedure, which is fully implicit for linear viscoelastic models, and semi-implicit for the quasi-linear case. We find that, in the correction phase, the discrete stress projection and plastic slip have the same form for all the considered models, although with different property and time-step dependent projection terms. A series of numerical experiments is carried out with analytical and reference solutions to demonstrate the convergence and computational cost of the proposed framework, which is shown to be at least first-order accurate for general loading conditions. Our numerical results demonstrate that the developed framework is more flexible, preserves the numerical accuracy of existing approaches while being more computationally tractable in the visco-plastic range due to a reduction of 50% in CPU time. Our formulation is especially suited for emerging applications of fractional calculus in bio-tissues that present the hallmark of multiple viscoelastic power-laws coupled with visco-plasticity.

Modelling viscoelastic relaxation mechanisms in thermorheologically complex Fe(III)-poly(acrylic acid) hydrogels

Mechanical properties of hydrogels with reversible transition metal-polymer crosslinks can be flexibly tuned depending on the dissociation kinetics of the metal bond. We use rheology to investigate the sol-gel transition of a Fe(III)-poly(acrylic acid) network with varying crosslinker content and model the corresponding mechanical relaxation at different stages of gelation. The system transitions from an unentangled chain regime to a crosslink dissociation dominated regime, where the relaxation is governed by two timescales with different activation energies. To account for the interplay of chain and crosslinker dynamics, a time-temperature-superposition procedure is introduced for both processes separately, thus separating the dynamic processes in these thermorheologically complex dynamic networks. The activation energy of chain relaxation remains unchanged whether or not the chain participates in the network. To model contributions to the dynamic modulus of each process, we combine concepts from fractional viscoelasticity with a generalized Maxwell model, which describes the dynamics of an unentangled chain solution with reversible crosslinks. This allows us to quantify the evolution of viscoelastic parameters in the course of gelation, where we find that the terminal relaxation time of the gels increases less than expected at high crosslinker contents. This result is attributed to a facilitated crosslink exchange mechanism and a lower pH of the gel matrix.

Fractional Calculus Approach to Reproduce Material Viscoelastic Behavior, including the Time–Temperature Superposition Phenomenon

The design of modern products and processes cannot prescind from the usage of viscoelastic materials that provide extreme design freedoms at relatively low cost. Correct and reliable modeling of these materials allows effective use that involves the design, maintenance, and monitoring phase and the possibility of reuse and recycling. Fractional models are becoming more and more popular in the reproduction of viscoelastic phenomena because of their capability to describe the behavior of such materials using a limited number of parameters with an acceptable accuracy over a vast range of excitation frequencies. A particularly reliable model parametrization procedure, using the poles–zeros formulation, allows researchers to considerably reduce the computational cost of the calibration process and avoid convergence issues typically occurring for rheological models. The aim of the presented work is to demonstrate that the poles–zeros identification methodology can be employed not only to identify the viscoelastic master curves but also the material parameters characterizing the time–temperature superposition phenomenon. The proposed technique, starting from the data concerning the isothermal experimental curves, makes use of the fractional derivative generalized model to reconstruct the master curves in the frequency domain and correctly identify the coefficients of the WLF function. To validate the methodology, three different viscoelastic materials have been employed, highlighting the potential of the material parameters’ global identification. Furthermore, the paper points out a further possibility to employ only a limited number of the experimental curves to feed the identification methodology and predict the complete viscoelastic material behavior.

Measuring biological materials mechanics with atomic force microscopy - Determination of viscoelastic cell properties from stress relaxation experiments

Cells are complex, viscoelastic bodies. Their mechanical properties are defined by the arrangement of semiflexible cytoskeletal fibers, their crosslinking, and the active remodeling of the cytoskeletal network. Atomic force microscopy (AFM) is an often-used technique for the study of cell mechanics, enabling time- and frequency-dependent measurements with nanometer resolution. Cells exhibit time-dependent deformation when stress is applied. In this work, we have investigated the stress relaxation of HeLa cells when subjected to a constant strain. We have varied the applied force (1, 2, 4, and 8 nN) and pause time (1, 10, and 60 s) to check for common assumptions for the use of models of linear viscoelasticity. Then, we have applied three models (standard linear solid, five element Maxwell, power law rheology) to study their suitability to fit the datasets. We show that the five element Maxwell model captures the stress relaxation response the best while still retaining a low number of free variables. This work serves as an introduction and guide when performing stress relaxation experiments on soft matter using AFM. RESEARCH HIGHLIGHTS: Cells exhibit linear viscoelastic properties when subjected to stress relaxation measurements at the studied different forces and times. The stress relaxation is best described by a five element Maxwell model. All three used models capture a softening and fluidization of cells when disrupting actin filaments.

Unbiased retrieval of frequency-dependent mechanical properties from noisy time-dependent signals

The mechanical response of materials to dynamic loading is often quantified by the frequency-dependent complex modulus. Probing materials directly in the frequency domain faces technical challenges such as a limited range of frequencies, long measurement times, or small sample sizes. Furthermore, many biological samples, such as cells or tissues, can change their properties upon repetitive probing at different frequencies. Therefore, it is common practice to extract the material properties by fitting predefined mechanical models to measurements performed in the time domain. This practice, however, precludes the probing of unique and yet unexplored material properties. In this report, we demonstrate that the frequency-dependent complex modulus can be robustly retrieved in a model-independent manner directly from time-dependent stress-strain measurements. While applying a rolling average eliminates random noise and leads to a reliable complex modulus in the lower frequency range, a Fourier transform with a complex frequency helps to recover the material properties at high frequencies. Finally, by properly designing the probing procedure, the recovery of reliable mechanical properties can be extended to an even wider frequency range. Our approach can be used with many state-of-the-art experimental methods to interrogate the mechanical properties of biological and other complex materials.

Viscoelastic Behavior and Constitutive Relation of Heavy Crude Oils

Heavy crude oil exhibits very complex viscoelastic behaviors due to its complex composition of resins, asphaltenes, saturates, and aromatics. It has a great influence on oil production and transportation. In this work, the viscoelastic behaviors of three different heavy crude oils were measured using a rotational rheometer. In conclusion, all of these heavy crude oils display linear viscoelastic behaviors in the experimental range. The loss modulus (E″) of the three crude oils decreased as the experimental temperature increased, and the variation trends of the three crude oils were basically the same. However, the experimental temperature has almost no effect on the storage modulus (E'), which always retained a constant value of 0.4 Pa. Furthermore, the storage modulus (E') and loss modulus (E″) increase as the angular frequency increases. To describe the physical deformation characteristics of viscoelastic materials, the generalized Maxwell model and the fractional derivative Maxwell model are used to establish the constitutive relation of heavy crude oil. In conclusion, the generalized Maxwell model and the fractional derivative Maxwell model can predict the experimental results very well. All of the square of the correlation coefficient (R ²) values are greater than 0.95. However, the number of fitting parameters for the fractional derivative Maxwell model is less than that for the fourth-order generalized Maxwell model which can save the calculating time. Therefore, the fractional derivative Maxwell model is suggested to describe the viscoelastic behavior of heavy crude oil in industrial applications.

On modeling the multiscale mechanobiology of soft tissues: Challenges and progress

Tissues grow and remodel in response to mechanical cues, extracellular and intracellular signals experienced through various biological events, from the developing embryo to disease and aging. The macroscale response of soft tissues is typically nonlinear, viscoelastic anisotropic, and often emerges from the hierarchical structure of tissues, primarily their biopolymer fiber networks at the microscale. The adaptation to mechanical cues is likewise a multiscale phenomenon. Cell mechanobiology, the ability of cells to transform mechanical inputs into chemical signaling inside the cell, and subsequent regulation of cellular behavior through intra- and inter-cellular signaling networks, is the key coupling at the microscale between the mechanical cues and the mechanical adaptation seen macroscopically. To fully understand mechanics of tissues in growth and remodeling as observed at the tissue level, multiscale models of tissue mechanobiology are essential. In this review, we summarize the state-of-the art modeling tools of soft tissues at both scales, the tissue level response, and the cell scale mechanobiology models. To help the interested reader become more familiar with these modeling frameworks, we also show representative examples. Our aim here is to bring together scientists from different disciplines and enable the future leap in multiscale modeling of tissue mechanobiology.

Suppression of mid-spatial-frequency waviness by a universal random tree-shaped path in robotic bonnet polishing

Mid-spatial-frequency (MSF) errors seriously damages the imaging performance of optical components. Path pattern is an important factor that affects the generation of MSF waviness in polishing. This paper proposes a versatile pseudo random tree-shaped path (RTSP) generation method imitating the growth rules of tree branch in nature, which can efficiently generate continuous, uniformly distributed and multi-directional paths on a specified surface. Furthermore, the effectiveness of RTSP to suppress MSF waviness is verified by numerical simulation and experimental results. Finally, the RTSP is applied to the polishing of Ti-6Al-4V titanium alloy mirror. Results indicate that grinding marks have been significantly removed, while no obvious MSF waviness is introduced.

Exploring cell and tissue mechanics with optical tweezers

Cellular and tissue biosystems emerge from the assembly of their constituent molecules and obtain a set of specific material properties. To measure these properties and understand how they influence cellular function is a central goal of mechanobiology. From a bottoms-up, physics or engineering point-of-view, such systems are a composition of basic mechanical elements. However, the sheer number and dynamic complexity of them, including active molecular machines and their emergent properties, makes it currently intractable to calculate how biosystems respond to forces. Because many diseases result from an aberrant mechanotransduction, it is thus essential to measure this response. Recent advances in the technology of optical tweezers have broadened their scope from single-molecule applications to measurements inside complex cellular environments, even within tissues and animals. Here, we summarize the basic optical trapping principles, implementations and calibration procedures that enable force measurements using optical tweezers directly inside cells of living animals, in combination with complementary techniques. We review their versatility to manipulate subcellular organelles and measure cellular frequency-dependent mechanics in the piconewton force range from microseconds to hours. As an outlook, we address future challenges to fully unlock the potential of optical tweezers for mechanobiology.

Constitutive Equations for Analyzing Stress Relaxation and Creep of Viscoelastic Materials Based on Standard Linear Solid Model Derived with Finite Loading Rate

The viscoelastic properties of materials such as polymers can be quantitatively evaluated by measuring and analyzing the viscoelastic behaviors such as stress relaxation and creep. The standard linear solid model is a classical and commonly used mathematical model for analyzing stress relaxation and creep behaviors. Traditionally, the constitutive equations for analyzing stress relaxation and creep behaviors based on the standard linear solid model are derived using the assumption that the loading is a step function, implying that the loading rate used in the loading process of stress relaxation and creep tests is infinite. Using such constitutive equations may cause significant errors in analyses since the loading rate must be finite (no matter how fast it is) in a real stress relaxation or creep experiment. The purpose of this paper is to introduce the constitutive equations for analyzing stress relaxation and creep behaviors based on the standard linear solid model derived with a finite loading rate. The finite element computational simulation results demonstrate that the constitutive equations derived with a finite loading rate can produce accurate results in the evaluation of all viscoelastic parameters regardless of the loading rate in most cases. It is recommended that the constitutive equations derived with a finite loading rate should replace the traditional ones derived with an infinite loading rate to analyze stress relaxation and creep behaviors for quantitatively evaluating the viscoelastic properties of materials.

Thermodynamic Restrictions in Linear Viscoelasticity

The thermodynamic consistency of linear viscoelastic models is investigated. First, the classical Boltzmann law of stress-strain is considered. The kernel (Boltzmann function) is shown to be consistent only if the half-range sine transform is negative definite. The existence of free-energy functionals is shown to place further restrictions. Next, the Boltzmann function is examined in the unbounded power law form. The consistency is found to hold if the stress functional involves the strain history, not the strain-rate history. The stress is next taken to be given by a fractional order derivative of the strain. In addition to the constitutive equations involving strain-rate histories, finding a free-energy functional, consistent with the second law, seems to be an open problem.

Weighted Fractional Calculus: A General Class of Operators

We conduct a formal study of a particular class of fractional operators, namely weighted fractional calculus, and its extension to the more general class known as weighted fractional calculus with respect to functions. We emphasise the importance of the conjugation relationships with the classical Riemann–Liouville fractional calculus, and use them to prove many fundamental properties of these operators. As examples, we consider special cases such as tempered, Hadamard-type, and Erdélyi–Kober operators. We also define appropriate modifications of the Laplace transform and convolution operations, and solve some ordinary differential equations in the setting of these general classes of operators.

Estimation of Tensile Modulus of a Thermoplastic Material from Dynamic Mechanical Analysis: Application to Polyamide 66

The mechanical properties of thermoplastic materials depend on temperature and strain rate. This study examined the development of a procedure to predict tensile moduli at different strain rates and temperatures, using experimental data from three-point-bending dynamic mechanical analysis (DMA). The method integrated different classical concepts of rheology to establish a closed formulation that will allow researchers save an important amount of time. Furthermore, it implied a significant decrease in the number of tests when compared to the commonly used procedure with a universal testing machine (UTM). The method was validated by means of a prediction of tensile moduli of polyamide PA66 in the linear elastic range, over a temperature range that included the glass-transition temperature. The method was applicable to thermo-rheologically simple materials under the hypotheses of isotropy, homogeneity, small deformations, and linear viscoelasticity. This method could be applicable to other thermoplastic materials, although it must be tested using these other materials to determine to what extent it can be applied reliably.

The effects of viscoelasticity on residual strain in aortic soft tissues

Residual stress is thought to play a critical role in modulating stress distributions in soft biological tissues and in maintaining the mechanobiological stress environment of cells. Residual stresses in arteries and other tissues are classically assessed through opening angle experiments, which demonstrate the continuous release of residual stresses over hours. These results are then assessed through nonlinear biomechanical models to provide estimates of the residual stresses in the intact state. Although well studied, these analyses typically focus on hyperelastic material models despite significant evidence of viscoelastic phenomena over both short and long timescales. In this work, we extended the state-of-the-art structural tensor model for arterial tissues from Holzapfel and Ogden for fractional viscoelasticity. Models were tuned to capture consistent levels of hysteresis observed in biaxial experiments, while also minimizing the fractional viscoelastic weighting and opening angles to correctly capture opening angle dynamics. Results suggest that a substantial portion of the human abdominal aorta is viscoelastic, but exhibits a low fractional order (i.e. more elastically). Additionally, a significantly larger opening angle in the fully unloaded state is necessary to produce comparable hysteresis in biaxial testing. As a consequence, conventional estimates of residual stress using hyperelastic approaches over-estimate their viscoelastic counterparts by a factor of 2. Thus, a viscoelastic approach, such as the one illustrated in this study, in combination with an additional source of rate-controlled viscoelastic data is necessary to accurately analyze the residual stress distribution in soft biological tissues. STATEMENT OF SIGNIFICANCE: Residual stress plays a crucial role in achieving a homeostatic stress environment in soft biological tissues. However, the analysis of residual stress typically focuses on hyperelastic material models despite evidence of viscoelastic behavior. This work is the first attempt at analyzing the effects of viscoelasticity on residual stress. The application of viscoelasticity was crucial for producing realistic opening dynamics in arteries. The overall residual stresses were estimated to be 50% less than those from using hyperelastic material models, while the opening angles were projected to be 25% more than those measured after 16 hours, suggesting underestimated residual strain. This study highlights the importance viscoelasticity in the analysis of residual stress even in weakly dissipative materials like the human aorta.

A Note on an Integral Transformation for the Equivalence between a Fractional and Integer Order Diffusion Model

This note tackles the equivalence problem between the fractional and integer order diffusion models. Unlike existing approaches, the existence of a unique integral transformation mapping the solution of the integer order model to a solution of the fractional order model of α=1/2 is proven. Moreover, the corresponding inverse integral transformation is formally established to guarantee the equivalence and well-posedness of the solutions of these models. Finally, as an example, the solution of a fractional order diffusion model α=1/2, obtained through the solution of its integer order counterpart and the proposed transformation, is compared with the solution derived by using the Fourier transform.