Standard Linear Solid Research Articles

Vibration Analysis of Viscoelastic Timoshenko Cracked Beams with Massless Viscoelastic Rotational Spring Models

Based on the equivalent bending stiffness of the viscoelastic cracked beam with open cracks, the corresponding complex frequency characteristic equations of a Timoshenko viscoelastic cracked beam are obtained by using the method of separation of variables and the Laplace transform. The vibration characteristics of a viscoelastic Timoshenko cracked beams with the standard linear solid model and Kelvin-Voigt model are investigated. By numerical examples, the effects of the crack location, crack number, crack depth, and slenderness ratio on the vibration characteristics of the viscoelastic cracked beams are revealed.

Time-dependent plastic behavior of bacteria leading to rupture

A study of the mechanical response of bacteria is essential in designing an antibacterial surface for implants and food packaging applications. This research evaluated the mechanical response of Escherichia coli under different loading conditions. Indentation and prolonged creep tests were performed to understand their viscoelastic-plastic response. The results indicate that varying loading rates from 1 μm/s to 5 μm/s show an increase in modulus of 182% and 90%, calculated in the loading and unloading cycles, respectively, and a decrease in adhesion force by 42%. However, on varying loads from 5 nN to 25 nN, nominal change is observed in both modulus and adhesion force. The rupture curve at 100 nN load shows elastic and a small plastic deformation accompanied by a sharp peak indicating the cell wall rupture. The rupture force at the peak was found to be 34.38 ± 5.15 nN, irrespective of the loading rate, making it a failure criterion for bacteria rupture. The creep response of bacteria increases (for 6 s) and then remains constant (for 15 s) with time, indicating that a standard linear solid (SLS) model applies to this behavior. This work attempts to evaluate the mechanical properties of E. coli bacteria focusing on its rupture by contact killing mechanism.

Influence of partial saturation on bulk modulus and attenuation: the experiment and the theoretical model

SUMMARY Modelling the dispersion and attenuation of seismic waves in partially saturated rocks is important for quantitative interpretation of seismic data. To observe the partial saturation effects on modulus and attenuation, three sandstone samples with different porosities were used in the laboratory experiment. The samples were measured by ultrasonic testing system under different water saturation using drainage process. The experimental results show that the bulk modulus more rapidly increased at higher saturations (>80 per cent), and attenuation peaks could be observed. Then we develop a new partial saturation model based on Chapman’s theory and poroelastic theory. The new model is consistent with the Gassmann equation when the water saturation is 0 or 100 per cent. By analysing the characteristic frequency, the new model can be concluded in the form of a standard linear solid model. A comparison of the new model with the White model and the measured data is conducted. The results show that the model performed well at predicting the effective bulk modulus and attenuation of the rock with different water saturation. Finally, we discussed the impact of frequency, fluids and pore structures on the new model. The results reveal that the new model will be helpful in discussing partial saturation in rocks.

Energy balance for fractional anti-Zener and Zener models in terms of relaxation modulus and creep compliance

Considering the linear constitutive model containing fractional integrals and Riemann-Liouville fractional derivatives, the power per unit volume is expressed in time domain in terms of stored energy and dissipated power per unit volume. Relaxation modulus and creep compliance corresponding to fractional anti-Zener and Zener models are calculated and restrictions on model parameters narrowing thermodynamical constraints are posed in order to ensure relaxation modulus and creep compliance to be completely monotone and Bernstein function respectively, that a priori guarantee the positivity of stored energy and dissipated power per unit volume. Both relaxation modulus and creep compliance for model parameters obeying thermodynamical constraints, proved that can also be oscillatory functions with decreasing amplitude. Model used in numerical examples of relaxation modulus and creep compliance is also analyzed for the asymptotic behavior near the initial time instant and for large time.

Temperature and Frequency Dependence of the Dynamic Viscoelastic Properties of Silicone Rubber.

Temperature-frequency sweep tests were performed on silicone rubber to investigate the dynamic viscoelastic properties. The test results show that the viscoelasticity of silicone rubber presents significant temperature dependence and frequency dependence. The dynamic viscoelastic test curves at different temperatures can be shifted along the logarithmic frequency coordinate axis to construct smooth master curves at the reference temperature of 20 °C, covering a frequency range of 10 decades, which indicates thermorheological simplicity on a macro level and frequency temperature equivalence of the silicone rubber material in the experimental temperature range. The van Gurp-Palmen plot and Cole-Cole plot for the test data at various temperatures merge into a common curve, which further validates thermorheological simplicity. The temperature dependent shift factors of silicone rubber material were well characterized by the Williams-Landel-Ferry equation. Moreover, the fractional-order differential Kelvin (FDK) model, the fractional-order differential Zener (FDZ) model, and the improved fractional-order differential Zener (iFDZ) model were used to model the asymmetric loss factor master curve. The result shows that the iFDZ model is in good agreement with the test results, indicating that this model is suitable for describing the asymmetry of dynamic viscoelastic properties of silicone rubber.

Fractional derivative modeling for characterizing stress relaxation properties of Hami melon during drying.

The viscoelastic properties of food materials will change significantly while drying is in progress, which greatly influences the food deformation caused by drying. This study aims to predict the viscoelastic mechanical behavior of Hami melon during drying using the fractional derivative model. To characterize the relaxation characteristics, based on the finite difference method, an improved Grünwald-Letnikov fractional stress relaxation model is proposed to derive an approximate discrete numerical solution of the relaxation modulus by applying the time fractional calculus. The Laplace transform method is used to verify the obtained results, and the equivalence of the two methods is proved. In addition, the stress relaxation tests prove that the fractional derivative model has a better prediction effect on the stress relaxation behavior of viscoelastic food than classical Zener model. The significant correlations between the fractional order and the stiffness coefficient and the moisture content is also studied. Which is negative correlation and positive correlation respectively.

Human cornea thermo-viscoelastic behavior modelling using standard linear solid model

BackgroundCorneal biomechanics is of great interest to researchers recently. Clinical findings relate them to corneal diseases and to outcomes of refractive surgery. To have a solid understanding of corneal diseases’ progression, it is important to understand corneal biomechanics. Also, they are essential for better explaining outcomes of refractive surgeries and their undesired consequences. There is a difficulty for studying corneal biomechanics in-vivo and multiple limitations arise for ex-vivo studies. Hence mathematical modelling is considered as a proper solution to overcome such obstacles. Mathematical modelling of cornea in-vivo allows studying corneal viscoelasticity with taking into consideration all boundary conditions existing in real in-vivo situation.MethodsThree mathematical models are used to simulate corneal viscoelasticity and thermal behavior in two different loading situations: constant and transient loading. Two models of the three are used for viscoelasticity simulation which are Kelvin-Voigt and standard linear solid models. Also, temperature rise due to the ultrasound pressure push is calculated using bioheat transfer model for both the axial direction and as a 2D spatial map using the third model (standard linear solid model).ResultsViscoelasticity simulation results show that standard linear solid model is efficient for describing the viscoelastic behavior of human cornea in both loading conditions. Results show also that the deformation amplitude obtained from standard linear solid model is more reasonable for corneal soft-tissue deformation with respect to corresponding clinical findings than that obtained from Kelvin-Voigt model. Thermal behavior results estimated corneal temperature rise to be roughly 0.2 °C, which conforms with FDA regulations for soft tissue safety.ConclusionStandard Linear Solid (SLS) model is better describing the human corneal behavior in response to constant and transient load more efficiently. Temperature rise (TR) for the corneal tissue of about 0.2 °C is conforming with FDA regulations and even less than the FDA regulations for soft tissue safety.

Prediction on the Performance of Polymer-Based Mechanical Low-Pass Filters for High-G Accelerometers

A polymer-based mechanical low-pass filter(m-LPF) for high-g accelerometers makes it possible to remove high-frequency transient noises from acceleration signals, thus ensuring repeatable and reliable measurement on high-g acceleration. We establish a prediction model for performance of m-LPF by combining a fundamental vibration model with the fractional derivative standard linear solid(FD SLS) model describing the storage modulus and loss modulus of polymers. Here, the FD SLS model is modified to consider the effect of m-LPF shape factor (i.e., thickness) on storage modulus and loss modulus. The prediction accuracy is verified by comparing the displacement transmissibility(or cut-off frequency) estimated using our model with that measured from 3 kinds of polymers(polysulfide rubber(PSR), silicone rubber(SR), and polydimethylsiloxane(PDMS)). Our findings will contribute a significant growth of m-LPF for high-g accelerometers.

Topological Landau–Zener nanophotonic circuits

Topological edge states (TESs), arising from topologically nontrivial phases, provide a powerful toolkit for the architecture design of photonic integrated circuits, since they are highly robust and strongly localized at the boundaries of topological insulators. It is highly desirable to be able to control TES transport in photonic implementations. Enhancing the coupling between the TESs in a finite-size optical lattice is capable of exchanging light energy between the boundaries of a topological lattice, hence facilitating the flexible control of TES transport. However, existing strategies have paid little attention to enhancing the coupling effects between the TESs through the finite-size effect. Here, we establish a bridge linking the interaction between the TESs in a finite-size optical lattice using the Landau–Zener model so as to provide an alternative way to modulate/control the transport of topological modes. We experimentally demonstrate an edge-to-edge topological transport with high efficiency at telecommunication wavelengths in silicon waveguide lattices. Our results may power up various potential applications for integrated topological photonics.

Nonlinear harmonic vibrations of laminate plates with viscoelastic layers using refined zig-zag theory. Part 1 – Theoretical background

The paper is devoted to the theoretical formulation of the harmonic vibrations problem for laminate plates in the von Kàrmàn geometrically non-linear regime. The plate layers are modelled as viscoelastic using the fractional Zener material with the linear elastic material being a special case. The model of the material is formulated with the separation of the volumetric and deviatoric strain. The laminate plate kinematics is described using the refined zig-zag theory. The vibrations of the plate are formulated using the alternative exponential form with complex-conjugate amplitudes. The harmonic balance method and the time-averaging are applied to derive the amplitude equation of the problem in hand. The efficiency and correctness of the formulation is verified in the continuation paper Part 2 – numerical analysis.

Dynamic Characteristics of a Composite Beam with Viscoelastic Layers under Uncertain-But-Bounded Design Parameters

In this study, dynamic characteristics of a composite beam with uncertain design parameters are analyzed. Uncertain-but-bounded parameters change only within certain specified limits. This study uses interval analysis to investigate a composite beam with viscoelastic layers whose behavior is described using the fractional Zener model. In general, parameters describing both elastic and viscoelastic layers can be uncertain. Several methods have been studied to determine the lower and upper bounds of the dynamic characteristics of a structure. Among them, the vertex method is a comparative method in which the lower and upper bounds of the dynamic characteristics are approximated using the first- and second-order Taylor series expansion. An algorithm to determine the critical combination of uncertain design parameters has also been described. Numerical examples demonstrate the effectiveness of the presented methods and the possibility of applying them to the analysis of systems with numerous uncertain parameters and high uncertainties.

Measurements and estimation of frequency-dependent multi-scale viscoelastic damping properties of materials

The state-of-art viscoelastic property measurement and estimation methods at different frequencies/length scales inherently couple structural effects or sometimes lose accuracy for relevant parameters at smaller length scales in the material. We report a unified approach to measuring and estimating the frequency-dependent viscoelastic damping properties across different frequency scales independently of the specimen size effect and boundary conditions. The Dynamic Mechanical Analysis (DMA) method with three-point bending configuration is extended to higher-fidelity modeling in terms of time–frequency scales and material micro-mechanical scales together. The methodology demonstrates ways to identify the possible sources of undesired dynamics in the measurement process and eliminate those effects. It includes exciter tip contact dynamics in three-point bending configuration, anharmonic energy transfer to different frequencies (fractal bifurcation), and unmodelled artifacts or noise. Critically reviewing the standard DMA procedure with normal mode synthesis, this paper proposes a time–frequency Fourier spectral method that resolves a broader frequency range that extends to even acoustic/ultrasonic wave regimes of viscoelastic damping. An algorithm proposed in this paper with the advanced viscoelastic model with multi-parameters (standard linear solid model) estimates the fundamental material parameters at small length scales, which is more insightful to obtain a frequency-dependent correlation of material response to general dynamic loading. Stiffness and damping properties are estimated from measured data and validated for self-consistencies via different controls regarding static preload, specimen resonance, filtered noise, and model fidelity across the frequency range, and certain consistency and convergence properties of a series solution. The unified approach with a single test will be helpful in the design and evaluation of a wide variety of materials and their applications in vibration and acoustic problems. The proposed technique further demonstrates a novel way to extend low-frequency measurement to high-frequency response modeling.

An Incremental Contact Model for Rough Viscoelastic Solids

Viscoelastic dissipation is long concerned to be one of the origins of kinetic friction. In this study, the viscoelastic contact of rough solids is investigated by combining the principle of Lee and Radok and a recently developed incremental contact model. Evolutions of the real contact area and the average interfacial separation are obtained for a loading-unloading circle. For standard linear solids with Gaussian random rough surfaces, the effects of the viscoelastic reduction ratio and the loading rate on the contact responses are discussed. The viscosity of contacting solids would lead to the hysteresis of the mechanical responses to external excitation. Based on the relation between the average interfacial separation and the normal load, the viscoelastic energy dissipation is calculated and the role of surface morphology is addressed. Generally, the dissipated energy is proportional to the root-mean-square height of the rough surface, but decays for surfaces with larger root-mean-square slopes. The results of the current study are helpful for studying the friction between solids.

The strain gradient viscoelasticity full field solution of mode-III crack problem

The size and viscosity effects are noticeable at the micro-/nano scale. In the present work, the strain gradient viscoelastic solution of the mode-III crack in an infinite quasi-brittle advanced material is proposed based on the strain gradient viscoelasticity theory using the Wiener–Hopf method. The solutions to the gradient-dependent viscoelastic crack problem are obtained directly by using the correspondence principle between the strain gradient viscoelasticity and strain gradient elasticity in Maxwell’s standard linear solid model. In this model, the stress near the crack tip is time-dependent and size-dependent. Besides, the stress near the crack tip is more significant than that based on gradient elasticity theory. Compared with the elastic strain gradient effect, the viscous gradient effect makes the stress field at the crack tip harden. The location and the value of maximum stress change with time, which differs from the case in strain gradient elasticity theory. The time that the normalized stress takes to stabilize also changes with the distance from the crack tip. When the viscosity effect is neglected or time tends to infinity, the strain gradient viscoelasticity theory can be reduced to the classical strain gradient elasticity theory.

A comparative analysis between viscoacoustic forward and adjoint wave equations based on Maxwell, Kelvin-Voigt, and SLS rheological models

Most seismic imaging applications still consider the subsurface as a perfect acoustic or elastic medium, ignoring the viscosity of rocky materials. That viscosity converts part of the mechanical energy of seismic waves into heat, affecting the amplitude and phase of the wavefield when propagating through the physical medium. In this work, we derive, implement and analyze a set of forward and adjoint viscoacoustic equations based on the Maxwell, Kelvin-Voigt (KV), and standard linear solid (SLS) rheological models. These equations are described in the time–space domain allowing us to solve with the finite difference method. For this implementation, we used Devito - a domain-specific language (DSL) and code generation framework to design highly optimized finite difference kernels for use in inversion methods. We analyzed the attenuating behavior, comparing each equation’s dissipation and dispersion effects in these models, both forward modeling equations through comparisons between snapshots, seismograms, and traces, and adjoint equations through RTM images.

Refined nonlinear fractional derivative model of vehicle-track coupling dynamics

The coupled vehicle-track system (CVTS) dynamics have been extensively investigated for decades. However, the calculation accuracy of prevailing vehicle-track coupling models needs to be improved in the high frequency range due to the inappropriate model simplification and neglect of material nonlinearity. In this study, we propose a refined numerical model of the CVTS that considers the nonlinear properties of the railpads and primary suspension using the fraction derivative Zener model. Furthermore, we more realistically simulate the wheelset, rail and railpad configuration with the elastic axle, solid finite element and surface-support models, respectively, and improve the computation efficiency by employing the mode superposition method. The results demonstrate that the refined CVTS model is more accurate than the classical model in simulating vehicle-track coupling dynamics above 2 kHz. In particular, there are significant differences in the dynamic response of the elastic wheelset model compared to the rigid model over a broad frequency range, with an 11% difference in the bogie acceleration response at the first dominant frequency. When the railpads are modeled using the surface-support model, the rail acceleration differences exceed 41% near 1 kHz and 44% near 2650 Hz, compared to the point-support model. Additionally, the rail response at various locations across the rail cross section can be calculated using the finite element method in this refined model. Overall. the proposed CVTS model provides high accuracy and efficiency for random vibration analysis, especially in the high frequency domain.

Modified fractional-Zener model—Numerical application in modeling the behavior of asphalt mixtures

To seek a constitutive model that can both describe the viscoelastic response of asphalt mixtures well and be applied to finite-element simulation, a modified fractional-order Zener model (MFZM) was adopted from the mathematical expressions describing viscoelastic materials based on fractional calculus theory in this study. The MFZM was verified by the generalized fractional-Zener model, and the corresponding model configuration was provided. In this model, the dynamic response and relaxation-modulus function were derived. Based on the relaxation-modulus function, a numerical algorithm for the MFZM that can be implemented in a finite-element environment was proposed. The ability of the fractional-Zener model (FZM) and MFZM to characterize the dynamic response of asphalt mixtures was compared using dynamic modulus test. The dynamic modulus and semicircular bending (SCB) tests of the asphalt mixture were simulated using the proposed numerical algorithm in the software ABAQUS. The results demonstrate that, unlike FZM, MFZM can characterize the asymmetric characteristics of viscoelastic response of the asphalt mixture in the frequency domain. The simulation result of the dynamic modulus test agreed with the analytical solution of the MFZM in the frequency domain, which illustrated the availability of the proposed numerical method for the response of the asphalt mixture in the frequency domain. Moreover, the consistency between the actual experimental and virtual results of SCB proved the accuracy of the model and applicability of the proposed numerical method, allowing the model to be used in a finite-element environment.

Thermomechanical Characterization and Modeling of NiTi Shape Memory Alloy Coil Spring.

Today, shape memory alloys (SMAs) have important applications in several fields of science and engineering. This work reports the thermomechanical behavior of NiTi SMA coil springs. The thermomechanical characterization is approached starting from mechanical loading-unloading tests under different electric current intensities, from 0 to 2.5 A. In addition, the material is studied using dynamic mechanical analysis (DMA), which is used to evaluate the complex elastic modulus E* = E' - iE″, obtaining a viscoelastic response under isochronal conditions. This work further evaluates the damping capacity of NiTi SMA using tan δ, showing a maximum around 70 °C. These results are interpreted under the framework of fractional calculus, using the Fractional Zener Model (FZM). The fractional orders, between 0 and 1, reflect the atomic mobility of the NiTi SMA in the martensite (low-temperature) and austenite (high-temperature) phases. The present work compares the results obtained from using the FZM with a proposed phenomenological model, which requires few parameters for the description of the temperature-dependent storage modulus E'.

Analysis of the variable‐order fractional viscoelastic modeling with application to polymer materials

AbstractThe variable‐order fractional (VOF) operator has been extensively employed to depict complex physical phenomena of viscoelastic materials, during which the reasonable form of the proper variable‐order (VO) function will significantly affect the applicability and accuracy of proposed fractional model. To this regard, this work aims to provide a technical guidance for VO function selection in VOF viscoelastic modeling. In this paper, we develop a VOF model based on the classical Zener model, the suitable VO function form and the applicability of VOF model are investigated by means of characterizing the stress relaxation and creep recovery behaviors, in which the monotonicity of VO functions (decreasing or increasing) are discussed in detail. The results demonstrate that the stress relaxation and creep recovery responses have a realistic physical meaning only when the VOF rheological model with a monotonically decreasing VO function. Furthermore, three definitions of VOFD are also considered to determine the optimal VO function form and the reasonability of the proposed model is validated by fitting the experimental data of polymer matric materials. The analysis found that the decreasing exponential VO function exhibits good prospects in describing the time‐dependent behaviors, and it was found that the V2 and V3 definition types performed better. The results also show that the proposed model has unique advantage compared to the other existing viscoelastic models. Finally, the proposed VOF model is employed to predict the viscoelastic behavior to test the model's predictive ability.

Identification of the extended standard linear solid material model by means of experimental dynamical measurements

A novel algebraic procedure for the non-parametric identification of the material model by means of dynamical test measurements is proposed. An extended Standard Linear Solid (SLS) material model is taken into account to model the material linear visco-elastic behavior. It consists of the series arrangement of fractional Kelvin model elements adopting real parameters and integer and non-integer order time differential operators, and of hysteretic Kelvin model elements adopting complex parameters and integer order time differential operators. Hysteretic Kelvin model elements are introduced to take account of the material hysteretic behavior. The material E(j⋅ω) complex modulus, is analytically modeled as the ratio of pseudo polynomials (non-integer power terms) in the j⋅ω Fourier variable. A multi-step, iterative, material model identification technique is here proposed to identify the unknown polynomial coefficients and the non-integer exponent values starting from E(j⋅ω) material discrete estimates from input-output dynamical measurements made on a beam specimen at different ω frequency values. Computational, nonphysical SLS elements resulting from the application of the identification procedure can be found and eliminated, so that a low order optimal model result. Some results obtained by applying the proposed identification technique with real experimental measurements are shown and discussed.