Frequency-dependent Material Properties Research Articles

Experimental methods for the characterization of the frequency-dependent viscoelastic properties of soft materials

A characterization method based on Rayleigh wave propagation was developed for the quantification of the frequency-dependent viscoelastic properties of soft materials at high frequencies; i.e., up to 4 kHz. Planar harmonic surface waves were produced on the surface of silicone rubber samples. The phase and amplitude of the propagating waves were measured at different locations along the propagation direction, which allowed the calculation of the complex Rayleigh wavenumbers at each excitation frequency using a transfer function method. An inverse wave propagation problem was then solved to obtain the complex shear/elastic moduli from the measured wavenumbers. In a separate, related investigation, dynamic indentation tests using atomic force microscopy (AFM) were performed at frequencies up to 300 Hz. No systematic verification study is available for the AFM-based method, which can be used when the dimensions of the test samples are too small for other existing testing methods. The results obtained from the Rayleigh wave propagation and AFM-based indentation methods were compared with those from a well-established method, which involves the generation of standing longitudinal compression waves in rod-shaped test specimens. The results were cross validated and qualitatively confirmed theoretical expectations presented in the literature for the frequency-dependence of polymers.

Application of resonant acoustic spectroscopy to asphalt concrete beams for determination of the dynamic modulus

In this paper, a new application of resonant acoustic spectroscopy (RAS) is examined for constructing asphalt concrete mastercurves from seismic testing. The frequency-dependent material properties can be characterized from multiple modes of vibration through the use of RAS. Beam-shaped asphalt specimens are tested at multiple temperatures to determine the resonance frequencies of the specimens. The resonance frequencies are estimated by applying a small load impulse and measuring the resulting acceleration through the specimens. Using RAS, the material properties of the specimens are determined numerically using the measured resonance frequencies. The results presented show that the frequency-dependent dynamic modulus of the asphalt concrete specimens can be characterized using several modes of vibration at each testing temperature.

A finite element model using a unified formulation for the analysis of viscoelastic sandwich laminates

In this paper we present a layerwise finite element model for the analysis of sandwich laminated plates with a viscoelastic core and laminated anisotropic face layers. The stiffness and mass matrices of the element are obtained by Carrera’s Unified Formulation (CUF). The dynamic problem is solved in the frequency domain with viscoelastic frequency-dependent material properties for the core. The dynamic behaviour of the model is compared with solutions found in the literature, including experimental data.

Frequency-Dependent Substrate Characterization via an Iterative Pole Search Algorithm

The characterization of frequency-dependent material properties is an important issue in nowadays high-speed interconnect design. This letter presents a practical method to determine the complex permittivity of a substrate material, by combining measurements with simulations. A rational permittivity model is determined by searching for its poles and residues using an iterative optimization method. Its accuracy is verified by comparing coplanar waveguide measurements with simulations based on the new material model.

Multi-physics design and optimization of flexible matrix composite driveshafts

Flexible matrix composites (FMCs) consist of low modulus elastomers such as polyurethanes which are reinforced with high-stiffness continuous fibers such as carbon. This fiber–resin system is more compliant compared to typical rigid matrix composites and hence allows for higher design flexibility. Continuous, single-piece FMC driveshafts can be used for helicopter applications. In the present investigation, an optimization tool using a genetic algorithm approach is developed to determine the best combination of stacking sequence, number of plies and number of in-span bearings for a minimum-weight, spinning, misaligned FMC helicopter driveshaft. In order to gain more insight into designing driveshafts, various loading scenarios are analyzed and the effect of misalignment of the shaft is investigated. This is the first time that a self-heating analysis of a driveshaft with frequency- and temperature-dependent material properties is incorporated within a design optimization model. The analysis assures that the material does not overheat and that allowables are not exceeded. The challenge is that the analysis needs to address several physical processes such as self-heating in the presence of material damping, conduction and surface convection, ply-level stresses and strains, buckling and dynamic stability. Quasi-static and dynamic temperature- and frequency-dependent material properties for a carbon-polyurethane composite are embedded within the model. For two different helicopter drivelines, weight savings of about 20% are shown to be possible by replacing existing multi-segmented metallic drivelines with FMC drivelines.

Computational study of ultra-wideband wave propagation into the human chest

Ultra-wideband (UWB) has potential applications in a variety of medical areas such as implant wireless sensors, microwave hyperthermia, imaging and radar. The wave propagation inside the human body must be known for an efficient system design of these different e-health applications. Nevertheless, little is known about the propagation of UWB signals inside the human body despite the enormous research efforts devoted to characterising radio propagation around the body. Therefore in this study, the UWB wave propagation through the human chest is characterised by electromagnetic simulations. For this sake, a voxel model of the human body that incorporates the frequency-dependent material properties of the human tissues is used. The wave propagation for line-of-sight (LOS) scenarios in the frequency bands of 100-1000 MHz and 1-5 GHz are evaluated. Both vertical and horizontal wave polarisations are considered. The power radiographs of the received signals inside the human chest are presented, which display the frequency and polarisation-dependent variations. The spectral analysis of the received signals reveals that the frequency range of 100-250 MHz, offers the minimum propagation loss. This frequency range of 250-1000 MHz is depicted as a region with less variant loss against frequency. Furthermore, the power delay profile (PDP) and the root mean square delay spread of the in-body channels is presented for different depths inside the body. These insightful results further the understanding of the propagation behaviour of UWB signals inside the human body, which will ease the development of in-body e-health applications.

Characterization of frequency-dependent material properties of human liver and its pathologies using an impact hammer

The current methods for characterization of frequency-dependent material properties of human liver are very limited. In fact, there is almost no data available in the literature showing the variation in dynamic elastic modulus of healthy or diseased human liver as a function of excitation frequency. We show that frequency-dependent dynamic material properties of a whole human liver can be easily and efficiently characterized by an impact hammer. The procedure only involves a light impact force applied to the tested liver by a hand-held hammer. The results of our experiments conducted with 15 human livers harvested from the patients having some form of liver disease show that the proposed approach can successfully differentiate the level of fibrosis in human liver. We found that the storage moduli of the livers having no fibrosis (F0) and that of the cirrhotic livers (F4) varied from 10 to 20 kPa and 20 to 50 kPa for the frequency range of 0-80 Hz, respectively.

Characterisation by Inverse Techniques of Elastic, Viscoelastic and Piezoelectric Properties of Anisotropic Sandwich Adaptive Structures

In this article we present recent developments regarding parameter estimation in sandwich structures with viscoelastic frequency dependent core and elastic laminated skin layers, with piezoelectric patch sensors and actuators bonded to the exterior surfaces of the sandwich. The frequency dependent viscoelastic properties of the core material are modelled using fractional derivative models, with unknown parameters that are to be estimated by an inverse technique, using experimentally measured natural frequencies and associated modal loss factors. The inverse problem is formulated as a constrained minimisation problem, and gradient based optimization techniques are employed. Applications are presented and discussed, focused on the identification of viscoelastic frequency dependent core material properties.

A Viscoelastic Sandwich Finite Element Model for the Analysis of Passive, Active and Hybrid Structures

In this paper we present a finite element model for the analysis of active sandwich laminated plates with a viscoelastic core and laminated anisotropic face layers, as well as piezoelectric sensor and actuator layers. The model is formulated using a mixed layerwise approach, by considering a higher order shear deformation theory (HSDT) to represent the displacement field of the viscoelastic core and a first order shear deformation theory (FSDT) for the displacement field of the adjacent laminated anisotropic face layers and exterior piezoelectric layers. The dynamic problem is solved in the frequency domain with viscoelastic frequency dependent material properties for the core. Control laws are also implemented for the piezoelectric sensors and actuators. The model behaviour in dynamics is assessed with the few solutions found in the literature, including experimental data, and a laminated composite active sandwich application is proposed. In this numerical application, velocity feedback control law is implemented for active control, using co-located piezoelectric patch sensors and actuators.

Controlling the Casimir force via the electromagnetic properties of materials

The control of the Casimir force between two parallel plates can be achieved through adjusting the frequency-dependent electromagnetic properties of materials of the two plates. We show that, for different plate separations, the main contribution to the Casimir force comes from different frequency regions: For smaller (larger) separation, it comes from the higher (lower) frequency region. When the separation of the plates increases, the Casimir force can vary from attractive to repulsive and/or vice versa, by selecting the two plates with suitable electromagnetic properties. We discuss how a restoring Casimir force, which varies from repulsive to attractive by increasing the separation, can be realized and that the stable equilibrium is formed at zero Casimir force.

Seismoelectric response of heavy oil reservoirs: theory and numerical modelling

We consider a linear poroelastic material filled with a linear viscoelastic solvent like wet heavy oils (oil as opposed to water or brines is wetting the surfaces of the pores). We extend the electrokinetic theory in the frequency domain accounting for the relaxation effects associated with resonance of the viscoelastic fluid. The fluid is described by a generalized Maxwell rheology with a distribution of relaxation times given by a Cole–Cole distribution. We use the assumption that the charges of the diffuse layer are uniformly distributed in the pore space (Donnan model). The macroscopic constitutive equations of transport for the seepage velocity and the current density have the form of coupled Darcy and Ohm equations with frequency-dependent material properties. These equations are combined with an extended Frenkel–Biot model describing the deformation of the poroelastic material filled with the viscoelastic fluid. In the mechanical constitutive equations, the effective shear modulus is frequency dependent. An amplification of the seismoelectric conversion is expected in the frequency band where resonance of the generalized Maxwell fluid occurs. The seismic and seismoelectric equations are modelled using a finite element code with PML boundary conditions. We found that the DC-value of the streaming potential coupling coefficient is also very high. These results have applications regarding the development of new non-intrusive methods to characterize shallow heavy oil reservoirs in tar sands and DNAPL contaminant plumes in shallow aquifers.

Optimal design and parameter estimation of frequency dependent viscoelastic laminated sandwich composite plates

Recent developments in optimization and parameter estimation of frequency dependent passive damping of sandwich structures with viscoelastic core are presented in this paper. A finite element model for anisotropic laminated plate structures with viscoelastic frequency dependent core and laminated anisotropic face layers has been formulated, using a mixed layerwise approach, by considering a higher order shear deformation theory (HSDT) to represent the displacement field of the viscoelastic core, and a first order shear deformation theory (FSDT) for the displacement fields of adjacent laminated face layers. The complex modulus approach is used for the viscoelastic material behaviour, and the dynamic problem is solved in the frequency domain, using viscoelastic material data for the core, assuming fractional derivative constitutive models. Constrained optimization of passive damping is conducted for the maximisation of modal loss factors, using the Feasible Arc Interior Point Algorithm (FAIPA). Identification of the frequency dependent material properties of the sandwich core is conducted by estimating the parameters that define the fractional derivative constitutive model. Optimal design and parameter estimation applications in sandwich structures are presented and discussed.

Transient solution for poro-viscoacoustic wave propagation in double porosity media and its limitations

SUMMARY The analytical transient acoustic solution and dispersion characteristics for the double-porosity model are obtained over the whole frequency range for a homogeneous medium. The solution is also obtained by approximating the double porosity model with a uniform poro-viscoacoustic model and then constructing the transient response. The comparison between the results of the two models shows the likely validity and limitations of numerical solutions using a poro-viscoacoustic model to represent a double porosity medium in the heterogeneous case. Our calculations show that the dissipation by local mesoscopic flow of the double porosity model is very hard to fit over the entire frequency range by a single Zener element. However, since seismic exploration is normally restricted to a fairly narrow frequency band, this means that for frequency-dependent material properties, such as attenuation, the values around the centre frequency of the source will primarily determine the wave propagation characteristics. We choose the relaxation function that just approximates the dispersion behaviour of the double porosity model around the source centre frequency. It is shown that if the frequency is much lower than the peak attenuation frequency of the double porosity model, then wave propagation can be well described by the poro-viscoaoustic model with a single Zener element. For most water-filled sandstones having a double porosity structure, this holds true across the seismic frequency range. The transient solution for the heterogeneous double porosity medium is numerically obtained by a time splitting and Fourier pseudo-spectral staggered-grid method. As illustrative examples, the 2-D wavefield in a two-layer, water-saturated double porosity model are approximated by poro-viscoacoustic and poro-viscoelastic methods, respectively.

Free and forced vibrations of a tyre using a wave/finite element approach

Free and forced vibrations of a tyre are predicted using a wave/finite element (WFE) approach. A short circumferential segment of the tyre is modelled using conventional finite element (FE) methods, a periodicity condition applied and the mass and stiffness matrices post-processed to yield wave properties. Since conventional FE methods are used, commercial FE packages and existing element libraries can be utilised. An eigenvalue problem is formulated in terms of the transfer matrix of the segment. Zhong's method is used to improve numerical conditioning. The eigenvalues and eigenvectors give the wavenumbers and wave mode shapes, which in turn define transformations between the physical and wave domains. A method is described by which the frequency dependent material properties of the rubber components of the tyre can be included without the need to remesh the structure. Expressions for the forced response are developed which are numerically well-conditioned. Numerical results for a smooth tyre are presented. Dispersion curves for real, imaginary and complex wavenumbers are shown. The propagating waves are associated with various forms of motion of the tread supported by the stiffness of the side wall. Various dispersion phenomena are observed, including curve veering, non-zero cut-off and waves for which the phase velocity and the group velocity have opposite signs. Results for the forced response are compared with experimental measurements and good agreement is seen. The forced response is numerically determined for both finite area and point excitations. It is seen that the size of area of the excitation is particularly important at high frequencies. When the size of the excitation area is small enough compared to the tread thickness, the response at high frequencies becomes stiffness-like (reactive) and the effect of shear stiffness becomes important.

Damping Characteristics of Cylindrical Laminates with Viscoelastic Layer Considering Temperature- and Frequency-Dependence

The vibration and damping characteristics of the cylindrical hybrid panels with viscoelastic layers were investigated by using a full layerwise shell finite element method which can consider temperature and frequency dependent material properties. The present layerwise shell theory can accurately represent the zig-zag in-plane and out-of-plane displacements of multilayered hybrid structures and can fully consider the transverse shear and normal strains and the perfect cylindrical geometry. The remarkable differences of the frequency response functions of cylindrical hybrid panels including constrained layer damping and co-cured sandwiched models were observed. Present results show that the damping mechanics of hybrid panels with viscoelastic layers may be greatly affected by their temperature and frequency dependent material properties and that the present full layerwise theory can be used to accurately predict the vibration and damping characteristics of hybrid shells with viscoelastic layers.

Modelling Power Law Dependencies of Frequency Dependent AC Conductivity and Permittivity of Conductor-Relaxor Composites

Porous lead magnesium niobate-lead titanate (PMN-PT 90:10) relaxors were impregnated with water to provide a model conductor-insulator mixture, to study their power law frequency dependency of ac conductivity, permittivity and phase angle. Relaxor materials with a range of open porosity filled with water created composites with conductor volume fractions ranging from 8.2% to 22.2%. The use of a high relative permittivity PMN-PT (∼ 8000) enabled the power law dispersion to be observed at relatively low frequencies (∼ 2 kHz). Good agreement was obtained between experimental data and predicted results based on a logarithmic mixing rule with a strong correlation between the power law exponent and conductor-insulator fraction. The model and numerical methods presented are considered a simple approach to interpret and predict and the frequency dependent properties of materials which similar heterogeneity.

Measuring Frequency Dependent Material Properties by Use of N-Parameter Models

The measurement of material properties is an important task. Quantities such as Young's modulus and the shear modulus, Poisson's ratio and material damping must be known when dealing with noise problems in analytical and numerical calculations. By classical methods, often only a single number is obtained which is real and independent of frequency. Sometimes, however, this is not sufficient. If a material behaves visco-elastically, we can use N-parameter-models to describe its behaviour. Then, the aforesaid properties can be described as complex numbers and in dependence of frequency, often with just a fistful of parameters which are valid for a wide frequency range. This paper gives a short introduction into the theory of visco-elastic materials and presents three different measurement examples of parameter identification.

Under sleeper pads—Influence on dynamic train–track interaction

The influence of under sleeper pads (USP) on dynamic train–track interaction is investigated in a parameter study. Two numerical models that are valid in different frequency intervals are used. The time-domain model DIFF is used to study wheel–rail contact forces, rail bending moments, rail vibrations (displacements, velocities and accelerations), sleeper vibrations and loads on sleepers at high frequencies, here 50–3000 Hz. The frequency-domain model GroundVib is adopted to calculate sleeper vibrations at low frequencies. Frequency-dependent material properties of rail pads, USP and ballast/subgrade are accounted for by viscoelastic spring–damper models that are calibrated with respect to measured data. It is found that USP influence dynamic train–track interaction mainly in the frequency range 0–250 Hz. Similar sleeper responses are calculated by the two models, although some differences in magnitudes are present.

Design and Demonstration of an Infrared Meanderline Phase Retarder

We compare design and measurements for a single-layer meanderline quarter-wave phase retarder, operating across the wavelength range from 8 to 12 micrometers (25 to 37.5 THz) in the infrared. The structure was fabricated using direct-write electron-beam lithography. With measured frequency-dependent material properties incorporated into a periodic-moment-method model, reasonable agreement is obtained for the spectral dependence of axial ratio and phase delay. As expected from theory, the single-layer meanderline design has relatively low throughput (23%), but with extension to multiple-layer designs, the meanderline approach offers significant potential benefits as compared to conventional birefringent crystalline waveplates in terms of spectral bandwidth, angular bandwidth, and cost. Simple changes in the lithographic geometry will allow designs to be developed for specific phase retardations over specified frequency ranges in the infrared, terahertz, or millimeter-wave bands, where custom-designed waveplates are not commercially available.

Phase Characterization of Reflectarray Elements at Infrared

The feasibility of a square-patch reflectarray element design is demonstrated at a frequency of 28.3 THz in the infrared (10.6 micrometer free-space wavelength) for the first time. Fabrication of arrays of various patch sizes was performed using electron-beam lithography, and the reflected phase as a function of patch size was characterized using an infrared interferometer. A numerical model for the design of these reflectarray elements was developed incorporating measured values of frequency-dependent material properties, and a comparison of computed and measured phase shows close agreement.