Standard Linear Solid Research Articles

Aberrometric, Geometrical, and Biomechanical Characterization of Sound-Induced Vibrational Modes of the Living Human Cornea

Repeatable and reliable assessment of corneal biomechanics with spatial resolution remains a challenge. Vibrational Optical Computerized Tomography (V-OCT), based on sound-wave elastography, has made it possible to investigate the natural resonant modes of the cornea and obtain the elastic moduli non-invasively. This pilot study presents a characterization of four corneal vibrational modes from aberrometric, geometrical, and biomechanical approaches in the living human cornea of five healthy volunteers by combining a corneal sound-wave generator, dual Placido–Scheimpflug corneal imaging, and the Ocular Response Analyzer (ORA) devices. Sound-induced corneal wavefront aberration maps were reconstructed as a function of sound frequency and isolated from the natural state. While maps of low-order aberrations (LOA) revealed symmetric geometrical patterns, those corresponding to high-order aberrations (HOA) showed complex non-symmetric patterns. Corneal geometry was evaluated by reconstructing corneal elevation maps through biconical fitting, and the elastic and viscous components were calculated by applying the standard linear solid model to the ORA measurements. The results showed that sound-wave modulation can increase high-order corneal aberrations significantly. Two frequencies rendered the corneal shape more prolate (50 Hz) and oblate (150 Hz) with respect to the baseline, respectively. Finally, both the elastic and viscous properties are sensitive to sound-induced vibrational modes, which can also modulate the corneal stress-strain response. The cornea exhibits natural resonant modes influenced by its optical, structural, and biomechanical properties.

Mathematical Modelling of Viscoelastic Media Without Bulk Relaxation Via Fractional Calculus Approach

In the present paper, several viscoelastic models are studied for cases when time-dependent viscoelastic operators of Lamé’s parameters are represented in terms of the fractional derivative Kelvin–Voigt, Scott-Blair, Maxwell, and standard linear solid models. This is practically important since precisely these parameters define the velocities of longitudinal and transverse waves propagating in three-dimensional media. Using the algebra of dimensionless Rabotnov’s fractional exponential functions, time-dependent operators for Poisson’s ratios have been obtained and analysed. It is shown that materials described by some of such models are viscoelastic auxetics because the Poisson’s ratios of such materials are time-dependent operators which could take on both positive and negative magnitudes.

Exponential Stability of Higher Order Fractional Neutral Stochastic Differential Equation Via Integral Contractors

ABSTRACTThe well‐posedness results for mild solutions to the fractional neutral stochastic differential system with Rosenblatt process with Hurst index is discussed in this article. To demonstrate the results, the concept of bounded integral contractors is combined with the stochastic result and sequencing technique. In contrast to previous publications, we do not need to specify the induced inverse of the controllability operator to prove the stability results, and the relevant nonlinear function does not have to meet the Lipschitz condition. Furthermore, exponential stability results for neutral stochastic differential systems with Poisson jump have been established. Finally, an application to demonstrate the acquired results is discussed. We demonstrate the fractional Zener model for wave equation obeying the viscoelastic materials as a practical application of the system studied, which is a generalization of classical wave equation.

The effects of surface tension on spherical oscillatory indentation of viscoelastic materials

Abstract The oscillatory indentation has become an attractive approach to characterizing the viscoelastic properties of soft biological solids. However, the influences of surface tension on oscillatory responses are ignored, which might lead to an inaccurate measurement of mechanical properties. In this work, the influences of surface tension on spherical oscillatory indentation for viscoelastic materials are investigated through the finite element method. The viscoelasticity of solids is characterized by the standard linear solid model and a sinusoidal displacement is applied as the excitation signal. During an entire cycle at the steady state of oscillation, both the average value of contact radius and the dissipated energy decrease due to the presence of surface tension. For the oscillatory responses at various frequencies, the existence of surface tension results in an increase of average force but a decrease of phase angle. The force amplitude at low frequencies becomes higher when surface tension is considered. For the evaluation of the complex modulus, neglecting the surface tension would lead to a significant overestimation of storage modulus at low frequencies and an obvious underestimation of loss modulus when the normalized frequency approaches 1. Our results provide a comprehensive understanding of the effects of surface tension on the mechanical responses of oscillation and the determination of viscoelastic properties through oscillatory indentation.

A Pressure- and Frequency-Dependent Multiscale Model for Describing the Wave Propagation Characteristics of Fluid-Saturated Porous Media

Abstract The mechanism of wave propagation in fluid-saturated porous media is influenced by pressure and frequency. Pressure dependence is mainly dominated by the opening and closing of compliant and stiff pores in rocks, as well as nonlinear deformation respect to high-order elastic constants. Frequency dependence is mainly reflected in the dispersion and attenuation caused by wave-induced fluid flow (WIFF). Therefore, the propagation characteristics of seismic waves in subsurface rocks when pressure and frequency are coupled have broad practical significance, such as geofluid discrimination and in situ stress detection. A new equivalent elastic modulus applicable to fluid-saturated porous media has been established, which simultaneously considers the effects of pressure and WIFF. First, the dual-porosity model is incorporated to account for the changes in rock porosity under pressure and corresponding linear and nonlinear deformations. Then, based on the heterogeneity of rock at the mesocale and microscale, a unified pressure- and frequency-dependent elastic modulus over a wide frequency band is established using the Zener model. The wave equation of fluid-saturated porous media is constructed using the new model, and the pressure- and frequency-dependent phase velocities are derived. Rock physics and digital simulation experiments are applied to analyze the variation of elastic parameters and velocity with pressure and frequency. Comparison with experimental measurement data shows that the new model has higher accuracy than traditional models, especially in the low effective pressure region and the frequency band respect to seismic exploration.

A thermodynamically‐based fractional model combined viscoelastic‐viscoplastic‐ductile damage with application to fiber‐reinforced polymer composites

AbstractA thermodynamically‐based fractional viscoelastic‐viscoplastic‐damage constitutive model combined with continuous damage mechanics (CDM) theory was established, in order to describe the rate‐dependent nonlinear behavior of fiber‐reinforced polymer composites (FRPCs). The fractional Helmholtz free energy consists of four contributions: viscoelastic (VE), viscoplastic (VP), hardening and damage, in which the VE and VP parts are constructed by fractional Zener and Scott‐Blair (SB) element forms respectively. The constitutive equation is obtained through Helmholtz free energy for the fractional Zener model, and plastic flow and hardening evolution law are all derived in the process. The ductile damage, coupled to both VE and VP free energy parts, is introduced through fractional damage energy release rates to model the degradation of material properties. The corresponding strain energy release rate and dissipation contributions are also derived. The fractional implicit time integration algorithms of proposed model are presented. The model is applied to validate tests of FRPCs under various loading conditions. The model validation and comparison are presented by simulating experimental data and existing models in the literature. And the corresponding evolution of dissipated energy is discussed to further valid the characterization ability of the model.Highlights A thermodynamical fractional constitutive model was developed for FRPCs. The Helmholtz free‐energy potential for fractional Zener model is adopted. The physical significance of fractional order parameters is explored. Fractional implicit integration algorithm of proposed model is implemented. The validation and comparison of the model are presented under various loads.

Role of viscoelasticity in the adhesion of mushroom-shaped pillars

The contact behaviour of mushroom-shaped pillars has been extensively studied for their superior adhesive properties, often inspired by natural attachment systems observed in insects. Typically, pillars are modeled with linear elastic materials in the literature; in reality, the soft materials used for their fabrication exhibit a rate-dependent constitutive behaviour. Additionally, conventional models focus solely on the detachment phase of the pillar, overlooking the analysis of the attachment phase. As a result, they are unable to estimate the energy loss during a complete loading-unloading cycle.
This study investigates the role of viscoelasticity in the adhesion between a mushroom-shaped pillar and a rigid flat countersurface. Interactions at the interface are assumed to be governed by van der Waals forces, and the material is modeled using a standard linear solid model. Normal push and release contact cycles are simulated
at different approaching and retracting speeds.
Results reveal that, in the presence of an interfacial defect, a monotonically increasing trend in the pull-off force with pulling speed is observed. The corresponding change in the contact pressure distribution suggests a transition from short-range to long-range adhesion, corroborating recent experimental and theoretical investigations.
Moreover, the pull-off force remains invariant to the loading history due to our assumption of a flat-flat contact interface. Conversely, in the absence of defects and under the parameters used in this study, detachment occurs after reaching the theoretical contact strength, and the corresponding pull-off force is found to be rate
independent. Notably, the hysteretic loss exhibits a peak at intermediate detachment speeds, where viscous dissipation occurs, which holds true in both the presence and absence of a defect. However, the presence of a defect shifts the region where the majority of viscous dissipation takes place.

Experimental and numerical analysis of the vibrational behaviour of metallic structures filled with damping materials

This work focuses on characterizing and modeling the vibrational behavior of hollow metallic structures filled with damping material. Vibration tests are first carried on a hollow structure, then two filling materials are considered: a cast viscoelastic material and a granular material composed of epoxy and polyurethane beads. Modal parameters (natural frequencies, modal damping and mode shapes) are extracted from the Frequency Response Functions with experimental modal analysis tools. Damping ratios greater than 5~\% are measured. Given the difference in the nature of the filling materials, dissipation mechanisms are different. The effectiveness of a damping material depends on the type of structural modes (bending, torsion or wall mode) to be damped, and dissipation in filling materials cannot be modelled by a constant loss factor or viscous damping. Consequently, a finite element modeling of the structure filled with viscoelastic material is proposed. Numerical modal parameters of the structure are correlated with experimental measurements. By extension, a bead mix material law, based on a fractional Zener model, is proposed, whose parameters are obtained by inverse identification.

Characterization and modelling of the damping properties of composite structures based on flax fibers

In this paper, an experimental-numerical modelling is proposed to determine the damping properties of composite structures based on flax fibers. Assuming that these properties are due to a viscoelastic behavior of each layer and each direction of these structures, three viscoelastic models were considered : the Maxwell-type Zener (ZM) model, the generalized Maxwell (GM) model, and the fractional derivative Zener (ZF) model. As well known, the viscoelastic models involve complex modulus and frequency dependence. So, the free vibration problem of these composite structures is complex and non-linear one. This nonlinear problem is solved by an high order Newton method (HON method) which permits to determine damping properties (damped frequency and the structural loss factor). Using experimental tests, an identification method, based on the HON and the Nelder–Mead methods, is applied to identify the rheological parameters of the proposed viscoelastic models. The comparison between experimental and numerical results demonstrates the efficiency of the proposed experimental–numerical method : all parameters for the three models are successfully identified. From, the results, only the GM or ZF models seem to be suitable for representing the viscoelastic behavior of flax/epoxy composite structures.

A high-efficiency Q-compensated pure-viscoacoustic reverse time migration for tilted transversely isotropic media

The attenuation and anisotropy characteristics of real earth media give rise to amplitude loss and phase dispersion during seismic wave propagation. To address these effects on seismic imaging, viscoacoustic anisotropic wave equations expressed by the fractional Laplacian have been derived. However, the huge computational expense associated with multiple Fast Fourier transforms for solving these wave equations makes them unsuitable for industrial applications, especially in three dimensions. Therefore, we first derived a cost-effective pure-viscoacoustic wave equation expressed by the memory-variable in tilted transversely isotropic (TTI) media, based on the standard linear solid model. The newly derived wave equation featuring decoupled amplitude dissipation and phase dispersion terms, can be easily solved using the finite-difference method (FDM). Computational efficiency analyses demonstrate that wavefields simulated by our newly derived wave equation are more efficient compared to the previous pure-viscoacoustic TTI wave equations. The decoupling characteristics of the phase dispersion and amplitude dissipation of the proposed wave equation are illustrated in numerical tests. Additionally, we extend the newly derived wave equation to implement Q-compensated reverse time migration (RTM) in attenuating TTI media. Synthetic examples and field data test demonstrate that the proposed Q-compensated TTI RTM effectively migrate the effects of anisotropy and attenuation, providing high-quality imaging results.

Calculation of Curie Temperature and Carriers Effective Masses of (Ga, Mn)As Diluted Magnetic Semiconductor from the Eight-Band k.p Model

An important step toward the development of spintronic information processing, transfer, and storage devices is the discovery of a dilute magnetic semiconductor (DMS) that demonstrates carrier-mediated ferromagnetism and maintains its magnetic properties above room temperature. Despite considerable progress made in recent decades, the maximum Curie temperature (∼200 K) remains well below room temperature. Mndoped GaAs are among the various diluted magnetic semiconductor materials that have been studied and have emerged as one of the most promising contenders for technological usage. The key characteristic of GaMnAs is its ferromagnetic properties, which are critical for spintronic devices. The primary goal of this work is to compute two critical properties of GaMnAs: the effective masses and the Curie temperature. The 8-band k.p model put forth by Kane was utilized to compile the GaMnAs band parameters using a MATLAB program. Our calculations are based on Lowdin perturbation theory, where spin-orbit, sp-d exchange interaction, and biaxial strain are taken into account. We calculated the Curie temperature of the GaMnAs alloy, within the mean-field approach based on the Zener model. Our findings demonstrated a direct proportionality relationship between the Curie temperature and the hole concentration (p) in GaMnAs. Moreover, we looked into how the Mn content and strain affected the effective masses of electrons, heavy, and light holes that were calculated for various crystallographic directions in GaMnAs. The effective masses that were calculated were highly dependent on the Mn content, which ranged from 1% to 5%, and the strain, which varied from −2% (tensile strain) to 2% (compressive strain).

Vibration control of nonlinear viscoelastic systems with displacement delay feedback

In this study, the nonlinear vibration characteristics of a nonlinear Zener model under harmonic excitation, equipped with linear and nonlinear displacement delay feedback control, were investigated. The amplitude–frequency curve of the system’s main resonance was computed using the multi-scale method. Stability conditions for the system were then established based on the Lyapunov stability theory. The influence of several parameters on the system’s dynamic behavior was also analyzed, including time delay, displacement feedback gain coefficient, nonlinear term coefficient, damping coefficient, and external excitation. The findings revealed that the nonlinear displacement feedback gain coefficient was observed to exert a more substantial effect on the system’s amplitude than other factors. Further, the nonlinear displacement delay feedback gain coefficient and time-delay value diminish the amplitude of the vibration, leading all solutions to achieve a stable state. In instances with time-delay control, the impact of the main system parameters on system vibration was significantly diminished. The aim of this study is to provide a theoretical basis for the implementation of displacement delay controllers in nonlinear viscoelastic systems.

A Description of the Isothermal Ageing Creep Process in Polymethyl Methacrylate Using Fractional Differential Models.

Fractional differential viscoelastic models can describe complex material behaviours and fit experimental data well; however, the physical significance of model parameters is difficult to express. In this study, the fractional differential Maxwell, Kelvin, and Zener models were used to fit the short-term creep compliance curves of polymethyl methacrylate at different ageing times. The model fits were in good agreement with the experimental data. As the ageing time increased, the fractional differential Zener model showed a relative increase in the modulus parameter of the spring and a relative decrease in the modulus parameter reflecting the viscosity of the spring-pot, which indicated that physical ageing made the material more elastic. The relaxation time of the material increased, which indicated that the physical ageing reduced the free volume of the material, hindered the movement of molecules/segments, and increased the time required for the material to reach equilibrium. The fractional order of the model decreased, which reflected the phenomenon that physical ageing reduced the creep compliance of the material. Using the relaxation time as the time scale, the creep curves at different ageing times under the same stress level could be superimposed, naturally presenting the time-ageing time equivalence principle.

An efficient physics-dimension-based reduction method for computing frequency response functions of viscoelastically damped systems

The frequency response functions (FRFs) are of critical interest to dynamic problems, however, they suffer from computational challenges for viscoelastically damped systems. In this paper, a physics-space-based reduction method is proposed for predicting the FRFs of large-scale viscoelastically damped systems involving the standard linear solid model. A physics-dimension subspace is constructed based on original system matrices and viscoelastic parameters, which can be easily generated by using a recursive manner. A projection basis generation algorithm is then developed to generate a standard orthonormal basis within the physics-dimension subspace. With the help of the standard orthonormal basis and the moment-matching-based reduction method, a physics-space-based reduction method is proposed for efficiently predicting the FRFs of large-scale viscoelastically damped systems. Unlike the widely used state-space reduction method, the reduced system of the proposed method can preserve system’s physical structure so that the physical meaning can be captured. Using both theoretical and numerical analyses, the proposed physics-space-based method is more accurate and efficient than the state-space-based reduction method.

Stress and power as a response to harmonic excitation of a fractional anti‐Zener and Zener type viscoelastic body

AbstractThe stress as a response to strain prescribed as a harmonic excitation is examined in both transient and steady state regime for the viscoelastic body modeled by thermodynamically consistent fractional anti‐Zener and Zener models by the use of the Laplace transform method. Assuming strain as a sine function, the time evolution of power per unit volume, previously derived as a sum of time derivative of a conserved term, which represents the rate of change of stored energy, and a dissipative term, which represents dissipated power, is investigated when expressed through the relaxation modulus and creep compliance. Further, two forms of energy and two forms of dissipated power per unit volume are examined in order to see whether they coincide.

Investigation of Dynamic Viscoelastic Asymmetric Response of PA6 Film Based on Fractional Rheological Model.

Polyamide 6 (PA6) film as a typical viscoelastic material, satisfies the time-temperature superposition (TTS), and demonstrates obvious dynamic strain amplitude and frequency correlation under dynamic load. The investigation of the dynamic mechanical behavior of PA6 film is essential to ensure the safety of these materials in practical applications. In addition, dynamic mechanical property testing under conventional experimental conditions generally focuses on the short-term mechanical performance of materials. Therefore, the dynamic viscoelasticity of PA6 film was tested using a dynamic thermo-mechanical analyzer (DMA) in this study, and the complex modulus master curve was constructed based on time-temperature superposition (TTS) to realize the accelerated characterization of long-term mechanical properties. Furthermore, according to experimentally obtained asymmetric characteristics of the Cole-Cole diagram and the loss modulus master curve of the PA6 film, the parameter distribution of the fractional Zener model and the modified fractional Zener model were compared, and the asymmetric dynamic viscoelastic response of PA6 film under different conditions was systematically investigated using these models. The results indicate that the modified fractional Zener model can truly describe the dynamic asymmetric characteristics of PA6 film, verify the feasibility and advantages of the modified fractional rheological model, and provide some theoretical guidance for exploring the tensile rheological mechanism of PA6 film.

Establishment and evaluation of dynamic viscoelasticity constitutive model for asphalt mixtures based on PCA model: Utilizing coal-based synthetic natural gas slag and phosphogypsum whisker as substitute fillers

To promote the secondary utilization of solid waste in asphalt pavement and reduce resource waste and environmental damage, the fine solid waste powder from the secondary treatment of solid waste was used as a substitute filler to prepare asphalt mixtures. To more precisely characterize the effects of filler type and substitution ratio on the dynamic viscoelastic mechanical characteristics of asphalt mixtures, the dynamic modulus test was conducted on asphalt mixtures, while the dynamic modulus master curve of asphalt mixtures was established based on the time-temperature equivalent principle and the Sigmoidal function. The improved generalized Sigmoidal (IGS) model, five-parameter fractional Zener (FFZ) model, six-parameter fractional Zener (SFZ) model and ten-parameter fractional Zener (TFZ) model were used to construct the shrinkage frequency curves of different asphalt mixtures under different constitutive relationships, whereas the applicability of the constitutive models was further evaluated by the principal component analysis (PCA) model. The research results showed that the dynamic modulus and phase angle of the mixture exhibited a similar change rule with the variation of temperature and frequency, and the use of solid waste-based fillers could effectively reduce the effect of frequency variation on the viscoelasticity of the mixture. The dynamic modulus master curve constructed by Sigmoidal function could accurately describe the functional relationship between the dynamic modulus with loading frequency and temperature, and the variation trend of dynamic modulus in a wider frequency range was flattened with the increase in frequency, and this change rule was not affected by the substitution ratio of the filler, while the use of solid waste-based filler could reduce the dependence of the mixture on the temperature. Compared to several fractional constitutive models, the IGS model exhibited a higher fitting accuracy for the dynamic viscoelasticity of several mixtures, and with the use of solid waste-based fillers, the deviation of fitting curves gradually decreased. A comparative study showed that the fractional constitutive model was difficult to accurately describe the dynamic viscoelasticity of the mixture, and the IGS model was preferred to be recommended as a calculation model to investigate the constitutive relationship on the dynamic viscoelasticity of the mixture.

Dynamic viscoelastic properties of ultra-large particle size asphalt mixture based on fractional derivative constitutive model

The aim of this paper was to accurately grasp and precisely characterize the dynamic viscoelastic properties of Ultra-Large Particle Size Asphalt Mixture (LSAM-50), and fully explore the features and simplified forms of the generalized fractional derivative Zener (GFDZ) model. The constitutive relations of the GFDZ model with n Abel elements were first given, and a modified GFDZ (mGFDZ) model was proposed. Then the master curves of LSAM-50, AC-13 and AC-20 mixtures were constructed, and the expression effects of different models and the dynamic viscoelastic behaviors of different asphalt mixtures were compared. The results showed that LSAM-50 was a typical viscoelastic material similar to traditional mixture. Both dynamic modulus and energy storage modulus decreased with increasing temperature. The phase angle increased first and then decreased with temperature increasing, and the peak value appeared near 40℃ at low frequency. The loss modulus increased first and then decreased with the temperature increasing. When the loading frequency was 0.1 Hz∼25 Hz, the peak value appeared at 0℃∼20℃. The rutting factors of LSAM-50 became larger with temperature reduction and frequency enhancement. When the number of Abel elements reached 2 in the GFDZ model or reached 3 in the mGFDZ model, the model fitting effects would no longer change significantly. The 2-mGFDZ model with only 5 parameters fitted master curves data satisfactorily and the physical meanings of parameters were clear. Comparing among three mixtures, LSAM-50, with the lowest asphalt content, showed significantly superior elasticity and better deformation resistance at medium and high temperatures. At low temperatures, the viscosity behavior of LSAM-50 was slightly inferior, but it may have pronounced plate properties considering the effect of aggregate skeletonization. The research results would promote new pavement structural design and mechanical response analysis.

Vibration analysis of a composite sandwich plate with a viscoelastic auxetic core, FG-CNTRC interior, and MEE–FGP exterior face sheets

This research has been conducted to analyze the free vibration behavior of a five-layered composite plate with a viscoelastic auxetic core. The plate comprises a viscoelastic auxetic core layer, functionally graded carbon nanotube reinforced composite (FG-CNTRC) interior, and magneto-electro-elastic functionally graded porous (MEE–FGP) exterior skins, which is rested on Winkler–Pasternak foundation. According to the magnetic–electric boundary conditions and Maxwell equations, the electric and magnetic potentials of the plate are determined. The macro-mechanical properties of the auxetic core have been derived based on Gibson’s model; the three-parameter Zener model has also been utilized to describe its viscoelastic behavior. Additionally, the equations of motion of the plate are obtained and solved by using Reddy’s third-order shear deformation theory (TSDT) and the Galerkin method, respectively. Moreover, various aspects of the current research have been validated by comparing the numerical results with those reported in the literature section. Overall, in the numerical result section, the effects of different parameters and conditions such as geometrical parameters, position of the plate’s interfaces of layers (thickness of the core and face layers), various distribution patterns and volume fractions of both CNT and porosity, external work parameters of the MEE skins, boundary conditions, Winkler–Pasternak stiffness coefficients, the relaxation time, and other parameters of the viscoelastic core on the natural frequency and loss factor of the composite plate are represented.

A fractional Hawkes process model for earthquake aftershock sequences

Abstract A new type of Hawkes process, known as the fractional Hawkes Process (FHP), has been recently introduced. This process uses a Mittag-Leffler density as the kernel function which is asymptotically a power law and so similar to the Omori–Utsu law, suggesting the FHP may be an appropriate earthquake model. However, it is currently an unmarked point process meaning it is independent of an earthquake’s magnitude. We extend the existing FHP, by incorporating Utsu’s aftershock productivity law and a time-scaling parameter from the fractional Zener Model to a marked version so that it may better model earthquake aftershock sequences. We call this model the ‘Seismic Fractional Hawkes Process’ (SFHP). We then estimate parameters via maximum likelihood and provide evidence for these estimates being consistent and asymptotically normal via a simulation study. The SFHP is then compared to the epidemic type aftershock sequence and FHP models on four aftershock sequences from Southern California and New Zealand. While it is inconclusive if the seismic fractional Hawkes process performs better in a retrospective predictive performance experiment, it does perform favourably against both models in terms of information criteria and residual diagnostics especially when the aftershock clustering is stronger.