Frequency-dependent Viscoelastic Behavior Research Articles

Frequency and time dependent viscoelastic characterization of pediatric porcine brain tissue in compression.

Understanding the viscoelastic behavior of pediatric brain tissue is critical to interpret how external mechanical forces affect head injury in children. However, knowledge of the viscoelastic properties of pediatric brain tissue is limited, and this reduces the biofidelity of developed numeric simulations of the pediatric head in analysis of brain injury. Thus, it is essential to characterize the viscoelastic behavior of pediatric brain tissue in various loading conditions and to identify constitutive models. In this study, the pediatric porcine brain tissue was investigated in compression with determine the viscoelasticity under small and large strain, respectively. A range of frequencies between 0.1 and 40Hz was applied to determine frequency-dependent viscoelastic behavior via dynamic mechanical analysis, while brain samples were divided into three strain rate groups of 0.01/s, 1/s and 10/s for compression up to 0.3 strain level and stress relaxation to obtain time-dependent viscoelastic properties. At frequencies above 20Hz, the storage modulus did not increase, while the loss modulus increased continuously. With strain rate increasing from 0.01/s to 10/s, the mean stress at 0.1, 0.2 and 0.3 strain increased to approximate 6.8, 5.6 and 4.4 times, respectively. The brain compressive response was sensitive to strain rate and frequency. The characterization of brain tissue will be valuable for development of head protection systems and prediction of brain injury.

Vertical kinematic response of an end-bearing pipe pile in fractional viscoelastic unsaturated soil under vertically-incident P-waves

This work investigates the kinematic response of a pipe pile embedded in a fractional-order viscoelastic unsaturated soil stratum over a rigid base subjected to vertically-propagating seismic P-waves based on a continuum model. The fractional standard line solid model is introduced to the governing equation of unsaturated soil to refine the frequency-dependent viscoelastic behavior of the soil skeleton. The Rayleigh-Love rod theory is utilized to characterize the vertical motion of the pipe pile. A rigorous series form solution for the vertical kinematic response of the end-bearing pipe pile under vertically incident P-waves is yielded theoretically with boundary conditions of pile-soil system. The proposed solution is then verified by comparing with the numerical result of the reported solution. Based on the given rigorous analyzed solution, the effects of physical parameters in pile-soil system on the vertical kinematic response of the end-bearing pipe pile under harmonic P-waves are studied in both the frequency and time domains through calculation example and parametric analysis. Finally, relevant meaningful conclusions are included and indicate that soil parameters and pipe pile dimensions have significant effect on the vertical kinematic response of the pipe pile.

Field–Frequency-Dependent Non-linear Rheological Behavior of Magnetorheological Grease Under Large Amplitude Oscillatory Shear

This article investigates the influence of frequency on the field-dependent non-linear rheology of magnetorheological (MR) grease under large amplitude oscillatory shear (LAOS). First, the LAOS tests with different driving frequencies were conducted on MR grease at four magnetic fields, and the storage and loss moduli under the frequency of 0.1, 0.5, 1, and 5 Hz were compared to obtain an overall understanding of the frequency-dependent viscoelastic behavior of MR grease. Based on this, the three-dimensional (3D) Lissajous curves and decomposed stress curves under two typical frequencies were depicted to provide the non-linear elastic and viscous behavior. Finally, the elastic and viscous measures containing higher harmonics from Fourier transform (FT)-Chebyshev analysis were used to quantitatively interpret the influence of the frequency on the non-linear rheology of MR grease, namely, strain stiffening (softening) and shear thickening (thinning), under LAOS with different magnetic fields. It was found that, under the application of the magnetic field, the onset of the non-linear behavior of MR grease was frequency-dependent. However, when the shear strain amplitude increased in the post-yield region, the non-linear rheology of MRG-70 was not affected by the oscillatory frequency.

Homogenous shear modulus reconstruction at multiple frequencies using the DICE method

The non-invasive technique of Digital Image Correlation Elastography was used to estimate the viscoelastic properties of a tissue-mimicking silicone phantom. Harmonic motion fields were generated over a frequency range of 40–90 ​Hz and measured at the phantom surface by 4 stereoscopic camera pairs. The storage modulus (G’) and the loss modulus (G”) were successfully reconstructed from the measured surface displacement data. Low variations were found for repeated measurements and the frequency dependent viscoelastic behavior was highly consistent across multiple excitation waveforms. Power-law analysis of the two viscoelastic properties provided exponent terms consistent with tissue-like materials.

On the frequency dependence of viscoelastic material characterization with intermittent-contact dynamic atomic force microscopy: avoiding mischaracterization across large frequency ranges.

Atomic force microscopy (AFM) is a widely use technique to acquire topographical, mechanical, or electromagnetic properties of surfaces, as well as to induce surface modifications at the micrometer and nanometer scale. Viscoelastic materials, examples of which include many polymers and biological materials, are an important class of systems, the mechanical response of which depends on the rate of application of the stresses imparted by the AFM tip. The mechanical response of these materials thus depends strongly on the frequency at which the characterization is performed, so much so that important aspects of behavior may be missed if one chooses an arbitrary characterization frequency regardless of the materials properties. In this paper we present a linear viscoelastic analysis of intermittent-contact, nearly resonant dynamic AFM characterization of such materials, considering the possibility of multiple characteristic times. We describe some of the intricacies observed in their mechanical response and alert the reader about situations where mischaracterization may occur as a result of probing the material at frequency ranges or with probes that preclude observation of its viscoelastic behavior. While we do not offer a solution to the formidable problem of inverting the frequency-dependent viscoelastic behavior of a material from dynamic AFM observables, we suggest that a partial solution is offered by recently developed quasi-static force–distance characterization techniques, which incorporate viscoelastic models with multiple characteristic times and can help inform dynamic AFM characterization.

A Finite Element Model for the Vibration Analysis of Sandwich Beam with Frequency-Dependent Viscoelastic Material Core.

In this work, a finite element model was developed for vibration analysis of sandwich beam with a viscoelastic material core sandwiched between two elastic layers. The frequency-dependent viscoelastic dynamics of the sandwich beam were investigated by using finite element analysis and experimental validation. The stiffness and damping of the viscoelastic material core is frequency-dependent, which results in complex vibration modes of the sandwich beam system. A third order seven parameter Biot model was used to describe the frequency-dependent viscoelastic behavior, which was then incorporated with the finite elements of the sandwich beam. Considering the parameters identification, a strategy to determine the parameters of the Biot model has been outlined, and the curve fitting results closely follow the experiment. With identified model parameters, numerical simulations were carried out to predict the vibration and damping behavior in the first three vibration modes, and the results showed that the finite model presented here had good accuracy and efficiency in the specific frequency range of interest. The experimental testing on the viscoelastic sandwich beam validated the numerical predication. The experimental results also showed that the finite element modeling method of sandwich beams that was proposed was correct, simple and effective.

On the modal decoupling of linear mechanical systems with frequency-dependent viscoelastic behavior

Linear Multi-Degree of Freedom (MDOF) mechanical systems having frequency-dependent viscoelastic behaviors are often studied and modelled in frequency or Laplace domains. Indeed, once this modelling process is carried out, it is not generally possible to reduce the obtained MDOF damped mechanical system to a set of uncoupled damped modal oscillators apart from some special cases. In this paper a general procedure has been proposed to transform a coupled linear mechanical system having frequency-dependent viscoelastic characteristics to a set of independent damped modal oscillators. The procedure is based on a linear co-ordinate transformation procedure using matrices in real field only. The approach is exact and based on the solution of one associated eigenproblem for the case of linearly viscous damping. In the general case of frequency-dependent viscoelastic materials, the approach includes an iterative procedure solving local eigenproblems.Some numerical results are reported to show the capabilities of the proposed approach.

Finite element analysis and experimental study on dynamic properties of a composite beam with viscoelastic damping

This paper investigates the frequency dependent viscoelastic dynamics of a multifunctional composite structure from finite element analysis and experimental validation. The frequency-dependent behavior of the stiffness and damping of a viscoelastic material directly affects the system's modal frequencies and damping, and results in complex vibration modes and differences in the relative phase of vibration. A second order three parameter Golla–Hughes–McTavish (GHM) method and a second order three fields Anelastic Displacement Fields (ADF) approach are used to implement the viscoelastic material model, enabling the straightforward development of time domain and frequency domain finite elements, and describing the frequency dependent viscoelastic behavior. Considering the parameter identification a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Agreement between the curve fits using both the GHM and ADF and experiment is within 0.001 percent error. Continuing efforts are addressing the material modulus comparison of the GHM and the ADF model. There may be a theoretical difference between viscoelastic degrees of freedom at nodes and elements, but their numerical results are very close to each other in the specific frequency range of interest. With identified model parameters, numerical simulation is carried out to predict the damping behavior in its first two vibration modes. The experimental testing on the layered composite beam validates the numerical predication. Experimental results also show that elastic modulus measured from dynamic response yields more accurate results than static measurement, such as tensile testing, especially for elastomers.

Effectiveness of multilayer viscoelastic insulators to prevent occurrences of brake squeal: A numerical study

The purpose of this publication is to give an overview of the actual role of multi-layered viscoelastic parts, so called “shims”, to prevent squeal noises of automotive brake systems. Since shear stress is usually used to damp thin structures in their bending modes it is commonly believed to be the largest underwent by shims. To check this assumption and considering that stresses underwent by shims cannot be measured experimentally, the authors have computed them with the help of a detailed and realistic finite element model. Contrary to what shims manufacturers say, this study exhibits the fact that shims are almost uniquely solicited in their normal direction in brake systems. Secondly, the study focuses on the added damping and stiffening induced by the viscoelastic materials. In order to take into account these materials, a realistic frequency dependent viscoelastic behavior has been integrated in the simulations. Finally, the study shows certains eigenmodes for which the viscoelastic behavior of the shims reveals instabilities that would not exist without it. It is shown that this is due to coalescence phenomenon.

Dynamic mechanical analysis and dynamic infrared linear dichroism study of the frequency-dependent viscoelastic behavior of a poly(ester urethane)

Dynamic infrared linear dichroism (DIRLD) data have been collected as a function of strain modulation frequency in a simultaneous DIRLD-dynamic mechanical analysis experiment. The frequency range is limited, but the DIRLD data indicate some similarities to the temperature-dependent data. The frequency-dependent infrared data do not show the direct correlation with the mechanical data that was evident in the temperature-dependent study. However, the frequency-dependent infrared data do provide some molecular insight, with a similarity to the temperature-dependent results in the 1800–1320 cm −1 region, a quadrature signal increasing with increasing frequency in the 1320–1000 cm −1 region, and complex changes in the low wavenumber region that are interpreted as polymer chain responses as a function of frequency.

Viscoelastic properties of the human medial collateral ligament under longitudinal, transverse and shear loading

Ligament viscoelasticity controls viscous dissipation of energy and thus the potential for injury or catastrophic failure. Viscoelasticity under different loading conditions is likely related to the organization and anisotropy of the tissue. The objective of this study was to quantify the strain- and frequency-dependent viscoelastic behavior of the human medial collateral ligament (MCL) in tension along its longitudinal and transverse directions, and under shear along the fiber direction. The overall hypothesis was that human MCL would exhibit direction-dependent viscoelastic behavior, reflecting the composite structural organization of the tissue. Incremental stress relaxation testing was performed, followed by the application of small sinusoidal strain oscillations at three different equilibrium strain levels. The peak and equilibrium stress–strain curves for the longitudinal, transverse and shear tests demonstrate that the instantaneous and long-time stress–strain response of the tissue differs significantly between loading conditions of along-fiber stretch, cross-fiber stretch and along-fiber shear. The reduced relaxation curves demonstrated at least two relaxation times for all three test modes. Relaxation resulted in stresses that were 60–80% of the initial stress after 1000 s. Incremental stress relaxation proceeded faster at the lowest strain level for all three test configurations. Dynamic stiffness varied greatly with test mode and equilibrium strain level, and showed a modest but significant increase with frequency of applied strain oscillations for longitudinal and shear tests. Phase angle was unaffected by strain level (with exception of lowest strain level for longitudinal samples) but showed a significant increase with increasing strain oscillation frequency. There was no effect of test type on the phase angle. The increase in phase and thus energy dissipation at higher frequencies may protect the tissue from injury at faster loading rates. Results suggest that the long-time relaxation behavior and the short-time dynamic energy dissipation of ligament may be governed by different viscoelastic mechanisms, yet these mechanisms may affect tissue viscoelasticity similarly under different loading configurations.

Scanning probe-based frequency-dependent microrheology of polymer gels and biological cells.

A new scanning probe-based microrheology approach is used to quantify the frequency-dependent viscoelastic behavior of both fibroblast cells and polymer gels. The scanning probe shape was modified using polystyrene beads for a defined surface area nondestructively deforming the sample. An extended Hertz model is introduced to measure the frequency-dependent storage and loss moduli even for thin cell samples. Control measurements of the polyacrylamide gels compare well with conventional rheological data. The cells show a viscoelastic signature similar to in vitro actin gels.