The role of psychophysics in rheology

Generalized vs. fractional: a comparative analysis of Maxwell models applied to entangled polymer solutions

Fractional viscoelastic models provide an excellent description of rheological data for polymer systems with power-law behaviour. However, the physical interpretation of their model parameters, which carry fractional units of time, often remains elusive. We show that for poly(ethylene oxide) (PEO) solutions, the fractional Maxwell model (FMM) requires much fewer model parameters than the classical generalized Maxwell model for a good description of the data and that it can be applied consistently to solutions with varying degrees of viscoelasticity. Despite their fractional units, the parameters exhibit scaling laws similar to classical parameters as a function of polymer concentration.

An efficient computational scheme for solving coupled time-fractional Schrödinger equation via cubic B-spline functions

The time fractional Schrödinger equation contributes to our understanding of complex quantum systems, anomalous diffusion processes, and the application of fractional calculus in physics and cubic B-spline is a versatile tool in numerical analysis and computer graphics. This paper introduces a numerical method for solving the time fractional Schrödinger equation using B-spline functions and the Atangana-Baleanu fractional derivative. The proposed method employs a finite difference scheme to discretize the fractional derivative in time, while a θ-weighted scheme is used to discretize the space directions. The efficiency of the method is demonstrated through numerical results, and error norms are examined at various values of the non-integer parameter, temporal directions, and spatial directions.

The power of weak, transient interactions across biology: A paradigm of emergent behavior

A growing list of diverse biological systems and their equally diverse functionalities provides realizations of a paradigm of emergent behavior. In each of these biological systems, pervasive ensembles of weak, short-lived, spatially local interactions act autonomously to convey functionalities at larger spatial and temporal scales. In this article, a range of diverse systems and functionalities are presented in a cursory manner with literature citations for further details. Then two systems and their properties are discussed in more detail: yeast chromosome biology and human respiratory mucus.

Time-fractional Caputo derivative versus other integrodifferential operators in generalized Fokker-Planck and generalized Langevin equations

Fractional diffusion and Fokker-Planck equations are widely used tools to describe anomalous diffusion in a large variety of complex systems. The equivalent formulations in terms of Caputo or Riemann-Liouville fractional derivatives can be derived as continuum limits of continuous-time random walks and are associated with the Mittag-Leffler relaxation of Fourier modes, interpolating between a short-time stretched exponential and a long-time inverse power-law scaling. More recently, a number of other integrodifferential operators have been proposed, including the Caputo-Fabrizio and Atangana-Baleanu forms. Moreover, the conformable derivative has been introduced. We study here the dynamics of the associated generalized Fokker-Planck equations from the perspective of the moments, the time-averaged mean-squared displacements, and the autocovariance functions. We also study generalized Langevin equations based on these generalized operators. The differences between the Fokker-Planck and Langevin equations with different integrodifferential operators are discussed and compared with the dynamic behavior of established models of scaled Brownian motion and fractional Brownian motion. We demonstrate that the integrodifferential operators with exponential and Mittag-Leffler kernels are not suitable to be introduced to Fokker-Planck and Langevin equations for the physically relevant diffusion scenarios discussed in our paper. The conformable and Caputo Langevin equations are unveiled to share similar properties with scaled and fractional Brownian motion, respectively.

Complete Generalization of the Equations for the Stress-Strain Curves of Concrete under Uniaxial Compression

The existence of more than thirty stress-strain equations, including those proposed by the government regulations in many countries, seems to indicate that additional, unifying, and at the same time generalizing research is necessary for this subject. Many expressions can be found to set or determine the initial modulus of elasticity of concrete, i.e., the modulus of elasticity of concrete when no load has been applied to it. This work proposes a complete generalization of the equations based on scalar damage models, applicable to all types of concrete tested under uniaxial compression with any constant rate of stress or strain, although in no case can it be considered a constitutive model. We prefer to discuss an equation that models the shape of the stress-strain curve. Thus, the shape of this curve is studied here in the same way a forensic scientist would, which is why we could see this work as an autopsy carried out on the test specimen through the trace left in the plane σ-ε by the straining process up until its inevitable outcome. That is to say, we believe in a purely phenomenological approach. The results are compared with the data obtained experimentally by analyzing test specimens made using various mixed portions of cement, water, and aggregates.

Deformation constitutive model of subgrade soil under intermittent cyclic loading

During the actual operation of subways, high-speed railway, and other infrastructure, subgrade soil will be subjected to periodic vibration and intermittent loading, i.e., intermittent cyclic load. To study the deformation characteristics of subgrade soil under intermittent cyclic loading, the Newtonian dashpot in the generalized Kelvin model is replaced by the fractional Abel dashpot, a fractional generalized Kelvin model is established. Then, a nonlinear viscoplastic body model considering damage is proposed, which is connected in series with the improved fractional generalized Kelvin model, and the deformation constitutive model of soil under intermittent cyclic loading is obtained. Finally, the effectiveness and applicability of the model are analyzed, and the model is compared with other constitutive models. The results show that the constitutive model established in the paper can reflect the combined effect of confining pressure, frequency, dynamic stress amplitude and other factors on the cumulative plastic strain of soil, and can well characterize the development law of three types of cumulative plastic strain curves of soil: stable, critical, and failure. The fitting parameters [Formula: see text] and [Formula: see text] tend to decay and rise, respectively, with the increase in loading stage N, which reflects the rationality and accuracy of the model.

Unified transient creep constitutive model based on the crack evolution of micritic bioclastic limestone

The surrounding rock at the exit of the No. 1 drainage tunnel of the Artashi Water Conservancy Project is micritic bioclastic limestone with 55% bioclastic material. This rock underwent unpredictable large and time-dependent deformation during excavation. To date, the mechanical behaviour of this kind of rock has rarely been studied. In this study, traditional triaxial compression tests and multilevel creep tests were conducted on micritic bioclastic limestone, and the results clarified the instantaneous and time-dependent mechanical properties of the rock. Considering that the essence of rock failure is crack growth, the crack strain evolution properties were revealed in rock triaxial compression tests and multilevel creep tests. Based on triaxial compression tests, the evolution of axial cracks with increasing deviatoric stress ratio R_d (ratio of deviatoric stress to peak deviatoric stress) was observed, and an axial crack closure element and new crack growth element were proposed. To simulate the creep behaviour of a rock specimen, the relationship of the rock creep crack strain rate with R_d was studied. A creep crack element was created, and the creep crack strain evolution equation was obtained, which closely fit the experimental data. Combining the 4 element types (elastic element, crack closure element, crack growth element, and creep crack element), a unified transient creep constitutive model (Mo’s model) was proposed, which represented both the transient and time-dependent mechanical properties of the micritic bioclastic limestone.

A Variable-Order Dynamic Constitutive Model for Clay Based on the Fractional Calculus

To accurately describe the deformation characteristics of clay under long-term cyclic load, based on fractional calculus theory, elastoplastic theory and the basic element model, a variable-order fractional dynamic model designed to predict accumulative strain of clay was exhibited. Firstly, the cyclic load was separated into static and alternating load in accordance with cyclic load characteristics, and the strain of clay under static and alternating load was analyzed. Then, on the basis of the variable-order Abel dashpot model, rheological theory and elastoplastic theory, the expressions of the rheological constitutive model and strain response were both obtained. Finally, in combination with the undrained dynamic triaxial testing of Zhan Jiang clay and Tian Jin soft clay, a series of analyses was carried out on the effectiveness and parameter sensitivity of the model when subjected to long-term cyclic loading. By comparing the dynamic constitutive model with pre-existing models, the superiority of the dynamic constitutive model is revealed. The results show that the dynamic constitutive model can characterize properly the deformation characteristics of clay under the action of long-term cyclic loading, especially in its accelerating stage. The parameter sensitivity of the model exhibits a growing trend with the increment of loading duration.

Experimental investigation of the creep behaviour of remoulded loess under different levels of compactness

Generally, a loess high fill project will undergo a time-dependent deformation and settlement process for a long time after the initial fill. Understanding the creeping behaviour of compacted loess is an important part of determining the stability of a compacted loess foundation. To study the creep behaviour of remoulded loess under different levels of compactness, we performed triaxial shear and triaxial creep tests using Q₂ loess specimens obtained from the new district of Yan’an city. Based on laboratory test results, the triaxial shear and creep characteristics of remoulded loess under different levels compactness are summarised. The regularity of instantaneous strain, creep strain, total accumulated strain and initial shear modulus were analysed and the relationship between the compactness and long-term strength of remoulded loess is provided. It was found that the remoulded loess becomes harder and its long-term strength increases with an increase in compactness. Furthermore, we propose a new creep model (HD), based on the hardening-damage mechanism, and have derived one-dimensional (1D) and three-dimensional (3D) creep equations based on this new creep model. This new creep model is flexible enough to fit the typical creep test curves of remoulded loess, while perfectly describing the tertiary creep stage. Finally, the sensitivity of the HD creep model parameters was analysed; the results indicate that the parameters denoted as α, γ, and β significantly affect the morphological changes and various stage characteristics are represented by the creep curve.

Architectural Significance of the Seokguram Buddhist Grotto in Gyeongju (Korea)

The purpose of this article is to disclose the hidden architectural proportions and true nature of the Korean national treasure in Seokguram Grotto, Gyeongju. The authors compare its features with those of other ancient hypogeal or ashlar constructions with the intention of rediscovering its relevant configuration and latent structural properties in order to demonstrate its uniqueness. The methods employed in the research belong initially to architectural design and composition to advance in the later stages to the nuances of stone masonry, lighting effects and especially cohesive construction. In this discussion and thorough analysis diverse philosophical and scientific subtleties are brought to the surface. The results demonstrate significant potential thanks to recent architectural developments, such as Tadao Ando’s Buddha Hill in Hokkaido (2017) and the authors’ own proposal for a Buddhist monument.

Impulse response function for Brownian motion

Motivated from the central role of the mean-square displacement and its second time-derivative - that is the velocity autocorrelation function in the description of Brownian motion and its implications to microrheology, we revisit the physical meaning of the first time-derivative of the mean-square displacement of Brownian particles. By employing a rheological analogue for Brownian motion, we show that the time-derivative of the mean-square displacement of Brownian microspheres with mass m and radius R immersed in any linear, isotropic viscoelastic material is identical to , where h(t) is the impulse response function (strain history γ(t), due to an impulse stress τ(t) = δ(t - 0)) of a rheological network that is a parallel connection of the linear viscoelastic material with an inerter with distributed inertance . The impulse response function of the viscoelastic material-inerter parallel connection derived in this paper at the stress-strain level of the rheological analogue is essentially the response function of the Brownian particles expressed at the force-displacement level by Nishi et al. after making use of the fluctuation-dissipation theorem. By employing the viscoelastic material-inerter rheological analogue we derive the mean-square displacement and its time-derivatives of Brownian particles immersed in a viscoelastic material described with a Maxwell element connected in parallel with a dashpot and we show that for Brownian motion of microparticles immersed in such fluid-like materials, the impulse response function h(t) maintains a finite constant value in the long term.

Simultaneous Characterization of Relaxation, Creep, Dissipation, and Hysteresis by Fractional-Order Constitutive Models

We considered relaxation, creep, dissipation, and hysteresis resulting from a six-parameter fractional constitutive model and its particular cases. The storage modulus, loss modulus, and loss factor, as well as their characteristics based on the thermodynamic requirements, were investigated. It was proved that for the fractional Maxwell model, the storage modulus increases monotonically, while the loss modulus has symmetrical peaks for its curve against the logarithmic scale log(ω), and for the fractional Zener model, the storage modulus monotonically increases while the loss modulus and the loss factor have symmetrical peaks for their curves against the logarithmic scale log(ω). The peak values and corresponding stationary points were analytically given. The relaxation modulus and the creep compliance for the six-parameter fractional constitutive model were given in terms of the Mittag–Leffler functions. Finally, the stress–strain hysteresis loops were simulated by making use of the derived creep compliance for the fractional Zener model. These results show that the fractional constitutive models could characterize the relaxation, creep, dissipation, and hysteresis phenomena of viscoelastic bodies, and fractional orders α and β could be used to model real-world physical properties well.

Biofilm mechanics in an extremely acidic environment: microbiological significance

A variety of natural biofilms were collected from an extremely acidic environment at Río Tinto (Spain). In order to provide insights into the structure-function relationship, the microstructure of the biofilms was explored using low temperature scanning electron microscopy (LTSEM) in combination with rheological analysis. The creep-recovery experiment results have demonstrated the typical behaviour of viscoelastic materials that combine both elastic and viscous characters. The LTSEM visualization and rheological characterization of biofilms revealed that the network density increased in bacterial biofilms and was the lowest in protist Euglena biofilms. This means that, in the latter biofilms, a lower density of interactions exist, suggesting that the whole system experiences enhanced mobility under external mechanical stress. The samples with the highest dynamic moduli (Leptospirillum-Acidiphilium, Zygnemopsis, Chlorella and Cyanidium) have shown the typical strain thinning behaviour, whereas the Pinnularia and Euglena biofilms exhibited a viscous thickening reaction. The Zygnemopsis filamentous floating structure has the highest cohesive energy and has shown distinctive enhanced resilience and connectivity. This suggests that biofilms should be viewed as soft viscoelastic systems the properties of which are determined by the main organisms and their extracellular polymeric substances. The fractional Maxwell model has been found to explain the rheological behaviour of the observed complex quite well, particularly the power-law behaviour and the characteristic broad relaxation response of these systems.

Effect of magnetic field on electroosmotic flow of viscoelastic fluids in a microchannel

Electroosmotic flow is an efficient transportation technology driven by applying an external electric field across the microchannel, which has a great potential for future application. This work is presented to study the unsteady electroosmotic flow of viscoelastic fluids combined with a constant pressure gradient and a vertical magnetic field through a parallel plate microchannel. For the reason that the upper and bottom walls of the parallel plate microchannel in microfluidic devices can be made of different materials, this leads to different hydrophobic properties, asymmetric zeta wall potentials, and different slip boundary conditions. The Navier slip model with different slip coefficients at walls is considered. The generalized Maxwell fluid with fractional derivative is adopted for the constitutive equation of the fluid. The analytical and numerical solutions of velocity are derived by employing the integral transform method and finite difference method, respectively. Excellent agreement is found between the numerical solutions and analytical solutions. Finally, the effects of fractional parameter α , relaxation time λ , slip coefficients a and b , the ratio of wall zeta potentials R ξ , Hartmann number H a , and electrical field strength parameter S on velocity profiles are interpreted graphically in detail.

The Fractional Derivative of the Dirac Delta Function and Additional Results on the Inverse Laplace Transform of Irrational Functions

Motivated from studies on anomalous relaxation and diffusion, we show that the memory function M(t) of complex materials, that their creep compliance follows a power law, J(t)∼tq with q∈R+, is proportional to the fractional derivative of the Dirac delta function, dqδ(t−0)dtq with q∈R+. This leads to the finding that the inverse Laplace transform of sq for any q∈R+ is the fractional derivative of the Dirac delta function, dqδ(t−0)dtq. This result, in association with the convolution theorem, makes possible the calculation of the inverse Laplace transform of sqsα∓λ where α<q∈R+, which is the fractional derivative of order q of the Rabotnov function εα−1(±λ,t)=tα−1Eα,α(±λtα). The fractional derivative of order q∈R+ of the Rabotnov function, εα−1(±λ,t) produces singularities that are extracted with a finite number of fractional derivatives of the Dirac delta function depending on the strength of q in association with the recurrence formula of the two-parameter Mittag–Leffler function.

Theoretical Analysis of Fractional Viscoelastic Flow in Circular Pipes: General Solutions

Fractional calculus is a relatively old yet emerging field of mathematics with the widest range of engineering and biomedical applications. Despite being an incredibly powerful tool, it, however, requires promotion in the engineering community. Rheology is undoubtedly one of the fields where fractional calculus has become an integral part of cutting-edge research. There exists extensive literature on the theoretical, experimental, and numerical treatment of various fractional viscoelastic flows in constraint geometries. However, the general theoretical approach that unites several most commonly used models is missing. Here we present exact analytical solutions for fractional viscoelastic flow in a circular pipe. We find velocity profiles and shear stresses for fractional Maxwell, Kelvin–Voigt, Zener, Poynting–Thomson, and Burgers models. The dynamics of these quantities are studied with respect to normalized pipe radius, fractional orders, and elastic moduli ratio. Three different types of behavior are identified: monotonic increase, resonant, and aperiodic oscillations. The models developed are applicable in the widest material range and allow for the alteration of the balance between viscous and elastic properties of the materials.

Fractional viscoelastic models for power-law materials

Soft materials often exhibit a distinctive power-law viscoelastic response arising from broad distribution of time-scales present in their complex internal structure. A promising tool to accurately describe the rheological behaviour of soft materials is fractional calculus. However, its use in the scientific community remains limited due to the unusual notation and non-trivial properties of fractional operators. This review aims to provide a clear and accessible description of fractional viscoelastic models for a broad audience and to demonstrate the ability of these models to deliver a unified approach for the characterisation of power-law materials. The use of a consistent framework for the analysis of rheological data would help classify the empirical behaviours of soft and biological materials, and better understand their response.

Viscous-viscoelastic correspondence principle for Brownian motion

Motivated by the classical expressions of the mean-square displacement and the velocity autocorrelation function of Brownian particles either suspended in a Newtonian viscous fluid or trapped in a harmonic potential, we show that for all timescales the mean-square displacement of Brownian microspheres with mass m and radius R suspended in any linear, isotropic viscoelastic material is identical to the creep compliance of a linear mechanical network that is a parallel connection of the linear viscoelastic material with an inerter with distributed inertance m_{R}=m/6πR. The synthesis of this mechanical network leads to the statement of a viscous-viscoelastic correspondence principle for Brownian motion which simplifies appreciably the calculations of the mean-square displacement and the velocity autocorrelation function of Brownian particles suspended in viscoelastic materials where inertia effects are non-negligible at longer timescales. The viscous-viscoelastic correspondence principle established in this paper by introducing the concept of the inerter is equivalent to the viscous-viscoelastic analogy adopted by Mason and Weitz [T. G. Mason and D. A. Weitz, Phys. Rev. Lett. 74, 1250 (1995)10.1103/PhysRevLett.74.1250].

Double power-law viscoelastic relaxation of living cells encodes motility trends

Living cells are constantly exchanging momentum with their surroundings. So far, there is no consensus regarding how cells respond to such external stimuli, although it reveals much about their internal structures, motility as well as the emergence of disorders. Here, we report that twelve cell lines, ranging from healthy fibroblasts to cancer cells, hold a ubiquitous double power-law viscoelastic relaxation compatible with the fractional Kelvin-Voigt viscoelastic model. Atomic Force Microscopy measurements in time domain were employed to determine the mechanical parameters, namely, the fast and slow relaxation exponents, the crossover timescale between power law regimes, and the cell stiffness. These cell-dependent quantities show strong correlation with their collective migration and invasiveness properties. Beyond that, the crossover timescale sets the fastest timescale for cells to perform their biological functions.

A Unique Fractional Derivative Operator to Simulate All Dynamic Piezoceramic Dielectric Manifestations: From Aging to Frequency-Dependent Hysteresis

Ferroelectric materials are utilized in many applications ranging from capacitors to data storage. The hysteresis frequency dependence of ferroelectric materials has been well studied. However, the long-term dynamic behavior including aging has not been well documented due to the long time frame required to gather experimental data, but it is critical for understanding the lifespan of these materials in application. Previous work has shown that the hysteresis frequency dependence of dielectric properties can be accurately modeled in the time domain using fractional derivative operators applied on a large frequency bandwidth. Currently, the lowest frequencies tested have been restrained to the hysteresis cycle quasi-static threshold. Below this threshold, the hysteresis shape remains unchanged. This research expands the current knowledge by validating the use of fractional derivative operators in long-term aging models. The model data are experimentally validated using aged piezoelectric samples up to 10⁷ s. These results confirm that the low and high dynamic dielectric material behaviors are linked and can be consequently modeled using fractional derivative operators.

A unified rheological model for cells and cellularised materials

The mechanical response of single cells and tissues exhibits a broad distribution of time-scales that often gives rise to a distinctive power-law rheology. Such complex behaviour cannot be easily captured by traditional rheological approaches, making material characterisation and predictive modelling very challenging. Here, we present a novel model combining conventional viscoelastic elements with fractional calculus that successfully captures the macroscopic relaxation response of epithelial monolayers. The parameters extracted from the fitting of the relaxation modulus allow prediction of the response of the same material to slow stretch and creep, indicating that the model captured intrinsic material properties. Two characteristic times, derived from the model parameters, delimit different regimes in the materials response. We compared the response of tissues with the behaviour of single cells as well as intra and extra-cellular components, and linked the power-law behaviour of the epithelium to the dynamics of the cell cortex. Such a unified model for the mechanical response of biological materials provides a novel and robust mathematical approach to consistently analyse experimental data and uncover similarities and differences in reported behaviour across experimental methods and research groups. It also sets the foundations for more accurate computational models of tissue mechanics.

A unified rheological model for cells and cellularised materials

The mechanical response of single cells and tissues exhibits a broad distribution of time-scales that often gives rise to a distinctive power-law rheology. Such complex behaviour cannot be easily captured by traditional rheological approaches, making material characterisation and predictive modelling very challenging. Here, we present a novel model combining conventional viscoelastic elements with fractional calculus that successfully captures the macroscopic relaxation response of epithelial monolayers. The parameters extracted from the fitting of the relaxation modulus allow prediction of the response of the same material to slow stretch and creep, indicating that the model captured intrinsic material properties. Two characteristic times, derived from the model parameters, delimit different regimes in the materials response. We compared the response of tissues with the behaviour of single cells as well as intra and extra-cellular components, and linked the power-law behaviour of the epithelium to the dynamics of the cell cortex. Such a unified model for the mechanical response of biological materials provides a novel and robust mathematical approach to consistently analyse experimental data and uncover similarities and differences in reported behaviour across experimental methods and research groups. It also sets the foundations for more accurate computational models of tissue mechanics.

Viscoelastic properties of shock wave exposed brain tissue subjected to unconfined compression experiments

Traumatic brain injuries (TBI) affect millions of people each year. While research has been dedicated to determining the mechanical properties of the uninjured brain, there has been a lack of investigation on the mechanical properties of the brain after experiencing a primary blast-induced TBI. In this paper, whole porcine brains were exposed to a shock wave to simulate blast-induced TBI. First, ten (10) brains were subjected to unconfined compression experiments immediately following shock wave exposure. In addition, 22 brains exposed to a shock wave were placed in saline solution and refrigerated between 30 minutes and 6.0 hours before undergoing unconfined compression experiments. This study aimed to investigate the effect of a time delay on the viscoelastic properties in the event that an experiment cannot be completed immediately. Samples from both soaked and freshly extracted brains were subjected to compressive rates of 5, 50, and 500 mm/min during the unconfined compression experiments. The fractional Zener (FZ) viscoelastic model was applied to obtain the brain's material properties. The length of time in the solution statistically influenced three of the four FZ coefficients, E₀ (instantaneous elastic response), τ₀ (relaxation time), and α (fractional order). Further, the compressive rate statistically influenced τ₀ and α.

'Electrical viscosity' of piezoresistive sensors: Novel signal processing method, assessment of manufacturing quality, and proposal of an industrial standard

Piezoresistive sensors become increasingly important in wearable devices, specifically sensors printed with piezoresistive inks. The electrical viscosity of these sensors causes phenomena such as stress relaxation, creep and hysteresis. This compromises sensor calibration and leads to inaccurate results. When subjecting a sensor to loading and unloading, the subsequent calculation of the fractional time derivative of the conductance leads to the elimination of the hysteresis. The order of the fractional derivative equals the magnitude of the sensor's viscosity. In our study, the viscosity of several pressure sensors was determined with a material testing machine and ranged from 0.55% to 22.35%. We found that shunt-mode sensors are less viscous than through-mode sensors. In addition, if a viscous material is placed on top of the sensor, then its viscosity increases. This is important as some calibration devices for pressure sensor arrays use rubber bladders which are viscous. From the perspective of manufacturing, the variation of the viscosity across a sensor array accounts for the quality of the pressure sensor mat and should be kept to a minimum. An industry standard is proposed that calculates and reports the sensor viscosity based on loading and unloading of the sensor and calculating the fractional time derivative of the conductance, whose fractional order reduces the pressure - conductance hysteresis to zero and thereby corresponds to the sensor's viscosity.

Influence of saline solution absorption and compressive rate on the material properties of brain tissue

Traumatic brain injuries (TBI) affect millions of people each year and can result in long-term difficulties in thinking or focusing. Due to the number of people affected by these injuries, significant research has been dedicated to determining the mechanical properties of the brain using postmortem tissue from animals harvested within 24 h. The postmortem brain tissue is often stored in a solution until a rheological experiment is ready to begin. However, the effect of storage duration on the mechanical behavior of brain tissue is not understood. In this paper, postmortem porcine brains were placed in normal saline solution (0.9% NaCl) and refrigerated between 30 min and 6.5 h to allow the brain to absorb the solution. Afterwards, samples from both soaked and freshly extracted brains were subjected to unconfined compression tests at compressive rates of 5, 50, and 500 mm/min. The fractional Zener viscoelastic model was applied to obtain the brain's mechanical properties. While the results did not show a significant relationship between absorption and the long-term stiffness (E_∞), both the relaxation time (τ₀) and fractional order (α) were statistically influenced by both the length of time in the solution and compressive rate. Further, the instantaneous stiffness (E₀) was statistically influenced by the length of time in solution, though not the compressive rate.

The approximate constant Q and linearized reflection coefficients based on the generalized fractional wave equation

The fractional parameter in a generalized fractional model is set to control the degree of absorption. However, it does not have an explicit physical meaning, even though it may be estimated from seismic data. Therefore, it is necessary to establish a common reference, which is physically significant for the fractional parameter in model applications. In this paper, a reference is presented according to the constant Q model. The proposed reference can be used to analyze the fractional parameter in different value interval ranges. When the fractional parameter is small, the related absorptive mechanism is equivalent to a constant Q model. When the fractional parameter is large, it reveals an attenuation mechanism corresponding to a frequency-dependent Q. This analysis makes the fractional parameter more practical in other applications. The study also investigates how to derive the generalized linearized reflection coefficient with the fractional parameter for amplitude variation with offset/frequency. The linearized formulas are used to directly analyze the effects of the parameter contrast. They can also be used to directly estimate the related parameters in detail. According to the study, a known fractional parameter could be analyzed in practice. The synthetic results confirmed that the theory could extend the application of the generalized fractional wave equation.

Perception and measurement of food texture: Solid foods

There is still a gap between instrumental measurement and sensory evaluation because of the complexity of food texture in spite of many efforts. In sensory evaluation, the terms describing the texture should be well understood by panelists, which poses a problem of establishing lexicons and training panelists. In the instrumental measurement, more efforts are required to understand the large deformation and fracture behavior of foods. The texture profile analysis (TPA) proposed by Alina Szczesniak, Malcolm Bourne, and Sherman has been applied to many foods, and was useful to develop the understanding of textures. But sometimes confusion of the interpretation of TPA parameters appeared. Many new techniques have been introduced to quantify TPA parameters. Recent efforts to fill the gap between sensory evaluation and instrumental measurements, human measurements, or physiological measurements have been introduced. This endeavor is an effort of synthesizing the dentistry and biomedical approach, sensory and psychological approach, and material science approach, and therefore, the collaboration among these disciplines is necessary. This manuscript mainly discusses texture studies for solid foods.

PRACTICAL APPLICATIONS

To fill the gap between the sensory evaluation and the instrumental measurement of texture, it is necessary to examine the physical change of foods during the oral processing. This will give us the designing principle of palatable and safe foods.

Universal avalanche statistics and triggering close to failure in a mean-field model of rheological fracture

The hypothesis of critical failure relates the presence of an ultimate stability point in the structural constitutive equation of materials to a divergence of characteristic scales in the microscopic dynamics responsible for deformation. Avalanche models involving critical failure have determined common universality classes for stick-slip processes and fracture. However, not all empirical failure processes exhibit the trademarks of criticality. The rheological properties of materials introduce dissipation, usually reproduced in conceptual models as a hardening of the coarse grained elements of the system. Here, we investigate the effects of transient hardening on (i) the activity rate and (ii) the statistical properties of avalanches. We find the explicit representation of transient hardening in the presence of generalized viscoelasticity and solve the corresponding mean-field model of fracture. In the quasistatic limit, the accelerated energy release is invariant with respect to rheology and the avalanche propagation can be reinterpreted in terms of a stochastic counting process. A single universality class can be defined from such analogy, and all statistical properties depend only on the distance to criticality. We also prove that interevent correlations emerge due to the hardening-even in the quasistatic limit-that can be interpreted as "aftershocks" and "foreshocks."

Weak and Strong Gels and the Emergence of the Amorphous Solid State

Gels are amorphous solids whose macroscopic viscoelastic response derives from constraints in the material that serve to localize the constituent molecules or particles about their average positions in space. These constraints may either be local in nature, as in chemical cross-linking and direct physical associations, or non-local, as in case of topological "entanglement" interactions between highly extended fiber or sheet structures in the fluid. Either of these interactions, or both combined, can lead to "gelation" or "amorphous solidification". While gels are often considered to be inherently non-equilibrium materials, and correspondingly termed "soft glassy matter", this is not generally the case. For example, the formation of vulcanized rubbers by cross-linking macromolecules can be exactly described as a second order phase transition from an equilibrium fluid to an equilibrium solid state, and amorphous solidification also arises in diverse physical gels in which molecular and particle localization occurs predominantly through transient molecuar associations, or even topological interactions. As equilibrium, or near equilibrium systems, such gels can be expected to exhibit universal linear and non-linear viscoelastic properties, especially near the "critical" conditions at which the gel state first emerges. In particular, a power-law viscoelastic response is frequently observed in gel materials near their "gelation" or "amorphous solidification" transition. Another basic property of physical gels of both theoretical and practical interest is their response to large stresses at constant shear rate or under a fixed macrocopic strain. In particular, these materials are often quite sensitive to applied stresses that can cause the self-assembled structure to progressively break down under flow or deformation. This disintegration of gel structure can lead to "yield" of the gel material, i.e., a fluidization transition, followed by shear thinning of the resulting heterogeneous "jelly-like" fluid. When the stress is removed, however, the material can relax back to its former equilibrium gel state, i.e., gel rejuvenation. In constrast, a non-equilibrium material will simply change its form and properties in a way that depends on processing history. Physical gels are thus unique self-healing materials in which the existence of equilibrium ensures their eventual recovery. The existence of equilibrium also has implications for the nature of both the linear and non-linear rheological response of gel materials, and the present paper explores this phenomenon based on simple scaling arguments of the kind frequently used in describing phase transitions and the properties of polymer solutions.

Unexpected viscoelastic deformation of tight sandstone: Insights and predictions from the fractional Maxwell model

Tight gas is one important unconventional hydrocarbon resource that is stored in tight sandstone, whose mechanical property greatly influences the tight gas production process and is commonly believed to be simply elastic when designing the stimulation plan. However, the experimental evidence provided in this work surprisingly shows that tight sandstone can deform in a viscoelastic way. Such an unexpected observation poses a challenge in accurately modelling the deformation process. We solve this problem by adopting the fractional Maxwell model to successfully derive the constitutive equation of tight sandstone, based on which not only all the experimental data can be interpreted quantitatively, but also reasonable and consistent predictions as to tight sandstone’s long-term deformation behaviour can be made. We then investigate the typicality of our results in China’s Changqing oilfield, which is one major centre of tight gas production and where the rock samples for experiments are obtained. It is estimated that a non-negligible portion of 18% tight sandstone samples in this area will probably display viscoelasticity. Finally, our work implies that the mechanical properties of other materials may also need further scrutiny to possibly uncover any unexpected behaviour, overlooking which may result in misleading predictions.

Power-law hereditariness of hierarchical fractal bones

In this paper, the authors introduce a hierarchic fractal model to describe bone hereditariness. Indeed, experimental data of stress relaxation or creep functions obtained by compressive/tensile tests have been proved to be fit by power law with real exponent 0 ⩽ β ⩽1. The rheological behavior of the material has therefore been obtained, using the Boltzmann-Volterra superposition principle, in terms of real order integrals and derivatives (fractional-order calculus). It is shown that the power laws describing creep/relaxation of bone tissue may be obtained by introducing a fractal description of bone cross-section, and the Hausdorff dimension of the fractal geometry is then related to the exponent of the power law.

Power-law rheology in the bulk and at the interface: quasi-properties and fractional constitutive equations

Consumer products, such as foods, contain numerous polymeric and particulate additives that play critical roles in maintaining their stability, quality and function. The resulting materials exhibit complex bulk and interfacial rheological responses, and often display a distinctive power-law response under standard rheometric deformations. These power laws are not conveniently described using conventional rheological models, without the introduction of a large number of relaxation modes. We present a constitutive framework using fractional derivatives to model the power-law responses often observed experimentally. We first revisit the concept of quasi-properties and their connection to the fractional Maxwell model (FMM). Using Scott-Blair's original data, we demonstrate the ability of the FMM to capture the power-law response of ‘highly anomalous’ materials. We extend the FMM to describe the viscoelastic interfaces formed by bovine serum albumin and solutions of a common food stabilizer, Acacia gum. Fractional calculus allows us to model and compactly describe the measured frequency response of these interfaces in terms of their quasi-properties. Finally, we demonstrate the predictive ability of the FMM to quantitatively capture the behaviour of complex viscoelastic interfaces by combining the measured quasi-properties with the equation of motion for a complex fluid interface to describe the damped inertio-elastic oscillations that are observed experimentally.

The constitutive properties of the brain paraenchyma

Fractional models have proven to be very useful for studying viscoelastic materials. We consider the fractional Zener model (also called four-parameter model) to study both the relaxation function and creep compliance. The analytical results are compared with the known experimental results of the human brain tissue to obtain the best fit and brain mechanical parameters. The results are also compared to the non-fractional Zener model and four-parameter Burgers model, indicating that the four-parameter fractional model gives a substantially better fit for the all experimental data.

An evaluation of acrylic complete dentures using the discrimination of elastic bodies or viscous fluids

The discrimination test on elastic bodies and viscous fluids was carried out for 16 young dentists and 13 denture patients, because it is considered that dentures affect the ability to discriminate food. The materials used in this test were six thermoplastic polyurethane elastomers with compressive elastic moduli from 11.8 to 62.0 MPa and six silicone impression pastes with coefficients of viscosity from 1.77 x 10(-2) to 23.0 x 10(-2) MPa x s. The subjects (dentists and patients) discriminated the materials based on differences in elastic moduli or coefficients of viscosity by chewing the materials for 10 s. From these experiments, it became clear that the ability to discriminate between elastic bodies or viscous fluids upon chewing, of subjects who use complete-dentures in combination with their own teeth and either an upper or lower complete-denture (S/D) or who have a set of complete-dentures in both jaws (D/D) decreased considerably, compared with that of subjects who have their own teeth in upper and lower jaws (S/S). The ratios of S/D and D/D against S/S were 0.5 and 0.3, respectively. This indicates that with an increase in the area occupied by dentures in the oral cavity, the value of the differential threshold increased markedly and made food discrimination more difficult. In addition, discriminating viscous fluids was 2.4 to 3.1 times more difficult compared with the discrimination of elastic bodies, independent of denture placement.